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1
fiMMBWRlCp
n
I
George Bell & Sons*
C AMBRXDGE IIATHEIIATIOAL 8EBIE8. Crown 8vo.
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CAMBRIDOE MATHEMATICAL SERIES.
NOTES ON
EOULETTES AND GLISSETTES
CambtiUge:
PRINTED BT C. J. CLAT, M.A. AND SONS,
AT THE UNIVEBSITT PRESS.
nsi
ki<c,^%:cf^
i-s
/.
NOTES
ON
ROULETTES AND GLISSETTES.
BY
W? HrBESANT, D.Sc, F.R.S.,
FELIiOW OF ST JOHN's COLLEGE, CAMBBIDaE.
SECOND EDITION ENLARGED,
CAMBRIDGE: DEIGHTON, BELL AND CO.
LONDON: GEOBGE BELL AND SONS,
YORK STREET, COVENT GARDEN.
1890 ..-
A
A
t
I
PREFACE TO THE FIRST EDITION.
^ nriHE following pages contain the explanation of methods,
-*- and the investigation of formulsB,. which I have for some
time past found useful in the discussion of the curves pro-
duced by the rolling or sliding of one curve on another.
These methods and formulae are with a few exceptions
original, and, I believe, new; and my object has been to
present, from a geometrical point of view, solutions of the
various problems connected with Roulettes and Qlissettes.
I have ventured to introduce, and employ, the word Glissette,
as being co-expressive with Roulette, a word which has been
in use amongst mathematicians for a considerable time.
The formula of Art. (34) is of course well known; it
is given in Salmon's Higher Plane Curves, in Walton and
Campion's Solutions, in Jullien's Problems, in Bertrand's
Differential Catculus, and probably in many other books.
The theorem of Art. (37) was enunciated some years
ago, for the particular case of a conic, by Mr Wolstenholme,
and extended by myself to the case of any curve. I have
however recently found a reference to it in the NouveUes
Annates for June, 1869, from which it appears that it was
given by Steiner in an early number of the same journal.
391377
VI PREFACE.
For the incisive method of Art. (78) I am indebted to
Mr Ferrers.
It will be seen that the general formula of Art. (50) in-
cludes most of those which precede it, while it is itself
included in that of Art. (77), and that the theorem of Art.
(60) reduces all cases of motion in one plane to the cases of
Articles (50) or (77).
In a future tract I hope to produce some further deve-
lopments of the ideas which are here somewhat briefly
treated.
W. H. BESANT.
December, 1869.
PKEFACE TO THE SECOND EDITION.
These * Notes ' have been out of print for a long time,
and I have frequently been requested to produce a new
edition, but, until recently, I have not been able to find the
requisite time for the purpose of doing so.
I have made considerable additions to the text and the
examples, but, as these notes by no means constitute an
exhaustive treatise on the subject, I retain the original
title.
I am much indebted to Mr A. W. Flux, Fellow of
St John's College, for kind assistance in the revision of
proof sheets.
W. H. BESANT.
April, 1890.
MM
CONTENTS.
ARTICLES
Infinitesiinals 1 — 4
ROULETTES.
Ellipse and Cycloid 5 9
General Theorems 10—14
ninstrations I5
Epicycloids and Hypocycloids 16 27
Gnrvatore of ronlettes 28 — 36
Arcs and areas 37 — 41
Locus of centre of curvatnre 42 — 44
Envelope roulettes 45 — 56
Examples.
GLISSETTES.
Cartesian equations
67, 58
Instantaneous Centre .
59, 60
Illustrations
61—69
Holditch's Theorem .
70
Amsler*s Planimeter .
71
4 • i
viu
CONTENTS.
ABTICLBS
Oentrodes 72—77
Moving triangle 78 — 80
Angle sliding 81, 82
Examples.
General motion of a Rigid body.
Translation and Rotation .
Motion about a fixed point
Circle rolling on fixed circle
Composition of spins
Boiling sphere, axodes
Screws
Cylindroid
Screws
Boiling sphere, axodes
83, 84
85—87
88
89—91
92—94
96—100
101
102—106
107—111
MiSGELLANBOUS EXAMPLES.
PRELIMINARY REMARKS ON
INFINITESIMALS.
1. An infinitesimal is a quantity which, under certain
assigned conditions, vanishes compared with finite quantities.
If two infinitesimals vanish in a finite ratio to each other,
they, are said to be of the same order.
Thus, if 6 vanish, sin and are of the same order, as are
also sin m0 and tan n0.
If two infinitesimals, a and )3, are such that the ultimate
ratio of ^ to a* is finite, ^ is said to be of the second order
if a be of the first order.
Thus, 1 — cos 0y when vanishes, is of the second order
if ^ be of the first order.
And, generally, an infinitesimal which has, ultimately, a
finite ratio to the rth power of another is said to be of the
rth order if that other be of the first order.
The order of an infinitesimal is, d priori^ arbitrary and
conventional ; but, if any standard be fixed upon, the orders
of all others are determinate.
2. Consider figure (1), in which is the centre of a
circle, and AP a small arc ; FN, PL perpendiculars on OA
and on the tangent at Ay and Q the point in which OP pro-
duced meets AL.
B. R. 1
2 PRELIMINARY REMARKS ON INFINITESIMALS.
Then, if OA = a, and AOP = 0, it can be shewn by Trigo-
nometry that, when is indefinitely diminished,
AL _ PL 1 QL 1
■■■I
and
AP ' AP' 2a' AP" 2a«'
PQ--PL _ 1
AP" "4a»'
Therefore, if AP is an infinitesimal of the first order, AL
is of the first order, PL of the second, QL of the third, and
PQ - PL of the fourth.
3. If a is an infinitesimal of the first order,
Xd^ + fuj? : vc? :: \ : v, ultimately,
\a* + fia^ is of the second order :
and generally, it will be seen that the order of an infinitesimal
is not aflfected by the addition to it of an infinitesimal of any
higher order.
If AP" is a small arc of a curve, and AP an equal arc of
its circle of curvature at A ; then PP' is of the third order,
and therefore, so far as quantities of the second order are
concerned, P may be taken to be coincident with P.
4. liAPy AQ are two infinitesimal arcs, of the first order,
of two curves touching each other at J., and if APy AQ be
equal, or ultimately equal to each other, the distance PQ
is of the second order, and therefore, so far as quantities of
the first order are concerned, P and Q may be taken to be
coincident.
It will be seen that all the preceding theorems are con-
tained in, or deducible iBrom, the 7th and 11th lemmas of the
first section of the Principia.
Thus, from Lemma XL, if AP, AP' are two infinitesimal
arcs of a curve of the same order, and PL, P'L' the corre-
sponding perpendiculars,
PL : P'L' :: AP" : AP^.
ROULETTES.
5. When a curve rolls on a fixed curve any given point
in the plane of the rolling curve describes a certain curve,
which is called a roulette.
Under the same heading we shall also include the curves
enveloped by any given lines, straight or curved, which are
carried with the rolling curve.
In dealing with roulettes the following is a fundamental
theorem.
If a curve roll on a fixed curve, the line joining the
point of contact with any point Q in the plane of the rotting
curve is the normal to the path of Q.
For, as the curve rolls, the point of the curve, P, in con-
tact with the fixed curve, has no motion, and the whole area
is, at the instant, turning round P: hence the direction of
motion of Q, i. e. the tangent to its path, is at right angles
to QP, and QP is the normal. (See fig. 2.)
6. Centre of curvature of roulette.
If PP'j Pp are equal infinitesimal arcs of the fixed and
rolling curves, so that Qp rolls into the position Q'P' (as in
fig. 9), QP, Q'P are consecutive normals of the roulette, and
E, the point of intersection of these lines, produced if neces-
sary, is the centre of curvature at the point of the roulette.
7. We commence with two particular cases as illus-
trations.
1—2
4 ROULETTES.
If a circle roll on the inside of the circumference of a
circle of double its radius, any point in the area of the rolling
circle traces out an ellipse*.
Let G be the centre of the rolling circle, and E the point
of contact (fig. 3).
Then, if the circle meet in Q, a fixed radius of the fixed
circle, the angle ECQ is twice the angle EOA, and therefore
the arcs EQ, EA are equal.
Hence when the circles touch at A, the point Q of the
rolling circle coincides with -4, and the subsequent path of
Q is the diameter through A,
Let P be a given point in the given radius (7Q, and draw
RPN perpendicular to OA ;
then, OQE being a right angle, EQ is parallel to RP, and
therefore CR = CP, and uR is constant.
Also PN : RN :: PQ : OR;
therefore the locus of R being a circle, the locus of P is an
ellipse, whose semi-axes are
OC+CP Bud OC-CP.
8. Properties of the ellipse are deducible from this
construction.
Thus, the point E being the instantaneous centre, PE is
the normal to the ellipse at the point, and PT, perpendicular
to it, and therefore parallel to OF, is the tangent.
A circle can be drawn through EPQT, since EPT, EQT
are right angles ; but the circle through QPE clearly passes
through R,
therefore, the angle ORT is a right angle and
ON : OR :: OR : OT
or OKOT=OR',
a known property of the tangent.
* Appendix to Oeometrical Conies y first edition, 1869.
BOULETTES. 5
Again, if PF meet OQ in G, the angles PQO, PFQ are
equal, being on equal bases, EQ, OQf ;
/. PG : PQ :: PQ : PF
or PO.PF^Pqt^0R\
a known property of the normal.
9. To find the irUrinsic equation of a cycloid.
If the circle BPQ rolls along the straight line AP, the
diameter BQ originally coinciding with AO (fig. 4), the point
Q traces out a cycloid of which is the vertex.
QP is the normal at the point Q of the cycloid, and if Pp
is an elementary arc of the circle, Qp turns into the position
QP', so that Q'P is the consecutive normal, and the point
E is the centre of curvature.
P'p being of the second order of infinitesimals, the points
p and P' may be taken to be coincident, and if PCp = SOy
pQp = m
As the circle turns through the angle S0, Qp turns
through the same angle, and therefore QpQ^ = S0; hence it
follows that QE(^ = ^S0, and therefore, ultimately, Q^= 2QP.
If the arc QQ^ = Ss, we have
Ss = 2PQ . iSd = 2a cos | SB,
and, QR being the tangent Q, the angle <f> of defiection
= i2PQ = i^;
.*. Ss 5=3 4a cos <l)B(f>
and 5 = 4a sin <^,
measuring 8 and ^ from the point 0, and the tangent at 0.
10. -4. cwrue roUs on a straight line ; it is required to find
the roulette traced by any point Q.
Let the curve roll from to P, the point A passing
over the point 0. (See fig. 2.)
6 ROULETTES.
Taking as the origin, and OP as axis of x, let a?, y, be
co-ordinates of Q,
Then, if AQP = 0, QP==r,
and y = r sin QPJV = r -r- .
Hence, if the polar equation referred to the point Q,
r =f (0), is given, we have three equations from which r
and can be eliminated, and the resulting equation will be
the diflferential equation to the path of Q.
Or, if the arc AP (= s) be found in terms of 0j we may
employ the equations
x = s — r cos NPQ = s — r -r- ,
y = r am QPN = r" -^ ,
and the elimination of r and will give the equation in x
and y, to the path of Q.
If the rolling curve is given by the equation p =if{r\
we have
is the roulette.
.•. smce p = y,
11. The two theorems following will be found of great
use in the discussion of roulettes.
If <l) is the deflection of the tangent at any point P of a
curve from the tangent at a fixed point of the curve from
which the arc is measured, and if p is the perpendicular from
a fixed point (0) on the tangent at P, then
ROULETTES. 7
(1) The perpendicular from O on the normal, measured
in the same direction as the arc, is equal to-^if the curve is
concave to the point O, and is--^ifihe curve is convex to the
point O.
(2) The radium of curvature a^ P =p + -r^ , if the curve
is concave to the point O, and = — p — -r-^ , if the curve is
convex to the point 0.
In fig. 5, if PP' is an elemental arc of the curve,
and if the tangents at P and P* intersect in T, Y'TY=^B<f),
and KY' = TKS<I>, neglecting infinitesimals of the second
order.
But Sp = OF ^OY^^KY' to the first order ;
therefore ultimately dp = P Yd(f>,
when measured in the direction PP\
In fig. 6,
so that OZ = — 5~ , when measured in the direction P'P,
d<l>
and is therefore -tr iu the direction PP',
dip
In fig. 7,
S/7 = or-OF=-irF=-rirs0,
so that 0Z=-^>
OAp
measured in the direction PP",
8 ROULETTES.
And in fig. 8,
therefore 0Z==~ , measured in the direction P'P. and this
cUj)
— -7? in the direction PP\
d<l>
Again, in fig. 5,
and radius of curvature = jj, = P + "y^t •
In fig. 6, S8^P7-TY+T7'-FY' = -B.PY+ KY'
ds _ cPp
In fig. 1,B8=TY- PY+ FY' -TY'=>B.PY- KY'
.. . da d?p
80 that _=_p-^;
and in fig. 8,
i8^PY-TY+TY'-FY' = -h.PY-KY
^,B<f>-pS<l>;
,1 . ds d^p
The cases of figures 6 and 7 are sufficient for the
argument ; the other cases are given for fullness of illustra-
tion. The same results are obtained by analytical methods,
as in Todhunter's Integral Calculus, Art. 90.
ROULETTES.
12. If <^ is the inclination of the tangent, or the
normal, to any fixed direction, and if p is the perpendicular
from a fixed point on the tangent to a curve, the relation,
P=/(^X is called the tangential polar equation of the
curve*.
We may remark that if <^ is the inclination of the normal
to a fixed line, p and ^ are the polar coordinates of the point
F; so that, putting r and for p and ^, the polar equation
of the pedal curve is
r =f(0).
For instance, the polar equation of the pedal of an ellipse
with regard to its centre is
r« = a» cos'^ + 6»sin»^;
so that the tangential polar equation of an ellipse, referred
to its centre, is
p^ =: a' cos* <j> + V sin* <j>.
13. We may observe that the intrinsic equation is at
once obtained by integrating the equation
da _ d^p
the right-hand member being expressed in terms of 0.
For instance, if OF is the perpendicular from on the
tangent RQ of a cycloid, (fig. 4), OY = OR sin <^ and
therefore p = 2a6 sin ^ is the tangential polar equation of
a cycloid referred to its vertex.
Hence j;? = 4a cos <^, and « = 4a sin ^.
14. To find the tangential polar equation of the roulette
traced by a point
Let fall OY, OZ, perpendiculars on the tangent QT and
the normal QP (fig. 10).
* This title was suggested by Dr Ferrers. {Cambridge and Dublin
Mathematical Journal, 1855.)
10 ROULETTES.
Let OY^p, and YOT-=^ (f>.
Then p = r— OP cos <^,
= r — 5 cos <f>,
whence, having r and 8 in terms of ^, and tan kJ) being -j- ,
we can eliminate r and and get the relation between p
and <^.
Or, without finding the arc, we have
^ = 0^ = P^ tan^
= (r — jp) tan^,
and, eliminating between this equation and
tan<^ = r^,
we get the diflferential equation, in p and <^, of the roulette.
15. Ex. 1. To find the roulette traced by the foeus of
a parabola rolling on a straight line:
In this case -- = 1 + cos ^,
r
the point A (fig. 2), being the vertex of the curve, and
Q the focus ;
.*. tanQPJV=cot^ = ;7-,
and y = r sin QPN =
'dx\'
cos 2
(dxV _ a'
whence, by integration,
X X
y = I (6- + 6'").
, Tfiat is, the roulette is a catenary.
ROXJLETTES. 11
Ex. 2. The curve r = a versin rolls on its dads; re-
quired ike locus of its pole.
Taking the second system of equations of Art 10, and
observing that
QPN= g , and « = 4a (l — cos ^ ) ;
we find that
a? = 4a [1 — cos ^ j — a(l — cos 0) cos^
= 4a — 2a cos ^ (2 + sin" ^ J ,
and y = 2a sin' ^ ;
whence 4a - a: = >/(2a)* - y* {2 (2a)* + y%
16. K the roulette be given in terms of x and y, we can
at once find the rolling curve.
For, Art 10,
da
dx
p = y,HJidp=-r-r-.
Hence, if y=f(x) be the roulette, we can eliminate
X and y, and find the equation, in p and r, to the rolling
curve, referred to the carried point as origin.
Ex. 1. If the roulette be the catenary
c - --
dx c , c re
-j-= -, and p = r. - = — ,
ds y ^ y p
.*. p^ = rc,
that is, the rolling curve is a parabola.
12 BOULETTES.
Ex. 2. If the roulette be
y' = 4taai,
the involute of a circle.
Ex. 3. If the roulette be
— J- — = 1
a' ^ b* '
the curve is
^ a
an epicycloid (Art. 22).
Ex. 4. If the roulette be
the cuiTe is
17. Roulettes of circles.
The equations in a? and y, or in polar coordinates, of
the roulettes produced by circles rolling on straight lines or
on circles are at once obtained from figures. Thus, in the
case of the cycloid, if ON'=x, and PN'=y (fig. 4),
a? = a — a cos ^, y^^ad + asinO,
OS I
and therefore y = cb vers"^ - + J2ax — a?,
is the Cartesian equation of a cycloid.
Again a cardioid is the roulette produced by a point in
the circumference of a circle rolling on an equal circle, and
if P, the tracing point, starts from A (fig. 11), and if ON^a,
and PN=y, the cardioid is given by the equations
flj = 2a cos 5 — a cos 2^, y = 2a sin ^ — a sin 20,
BOniiETTES. 13
If AP = r, we at once get the polar equation of the
cardioid referred to its cusp,
r = 2a (1 - cos 0),
PQ being the diameter through P, the point Q traces out a
cardioid of which B is the cusp, and if BQ = r, its equation is
r «= 2a (1 + cos 0).
Again, since
BF = BE+EF^iaco3 + 2a(l-co8 0)=:2a,
the locus of -F is a circle, centre B and radius AB, and FP
is the tangent at F.
Hence it follows that the cardioid described by the point
P is the pedal with regard to A, of the circle, centre B and
radius BA,
18. Epicycloids and Hypocycloids,
An epicycloid is the curve traced by a point in the
circumference of a circle rolling outside a fixed circle.
A hypocycloid is the curve traced by a point in the
circumference of a circle rolling inside a fixed circle.
Thus for an epicycloid, if a and b are the radii of the
fixed and rolling circles (fig. 12), and if
AOP = 0, QCP = ^;
.-. a? = (a + 6) cos ^ — 6 cos — j— 0,
y = (a + 6) sin ^ — 6 sin —j— 0.
For a hypocycloid we obtain in the same manner
x=^{a — o) cos ^ + 6 cos — j-— 0,
y = (a — 6) sin 5 — J sin —7 — ft
14 ROULETTES.
19. The area swept over by the radius vector is most
easily found by help of these equations and the expression
which is at once obtained from figure (13) as follows.
If OF, OQ are consecutive radii of a curve, a;, y the
coordinates of P, and x + dx, y + dy, of Q, the elemental
triangle
OPQ = OQL - OPN^ PL - PQR
1 11
=^^{(c + dx)ijf.+ dy)-^xy''ydx-^dx.dy
= \(xdy^ydx\
and therefore the area swept over by the radius vector
For instance, in the case of a cycloid (fig. 4), the area
swept over by OQ from the vertex to the cusp
= ^j(QN.d.ON-ON.d.QN)
= i TK^ + sin 0) sin ^ - (1 - cos 0) (1 + cos 0)} d0
^ Jo
1 «
= 2,ra.
Adding Tra*, and doubling the result, we obtain the area
between the curve and the straight line joining two
consecutive cusps.
20. The roulettes traced by the centres of circles rolling
on curves belong to the class of parallel curves.
If 6 is the radius of the circle, and it x, y are the
coordinates of its centre, and x', y' of the point of contact,
aj = a/ + 6 cos ^ and y = ^ + 6 sin ^,
where ^ is the inclination of the normal to the axis of x.
BOULETTES. 15
Or, if the given curve be p =/(^), the parallel is
Thus the parallels of a parabola referred to its focus and
of an ellipse referred to its centre are respectively
p = a sec 4- d, and p=^ Ja* cos* ^ -f- 6* sin" + d.
21. To find the tangential polar equation, and the
intrinsic equation, of an epicycloid.
In fig. (12), let OA ^a,CP = 6, and AOP = 0.
Then, if Q is the point tracing out the epicycloid, QD is
the tangent at Q, and taking <^ as the deflection of the
tangent at Q from the tangent at A, <f> is the inclination of
OT the perpendicular p on the tangent at Q, to the fixed
line OB at right angles to OA.
From the figure,
aod .*. p = (a + 26) sin ^ .
^^* *=-2r"^'
is the tangential polar equation of the epicycloid.
Hence ^ = (a + 26) (l - ^-^^.) sin ^
atf)
26
46 (a + 6) . ad>
— — ^^ ^ sm ^
a + 26 a + 26"
and therefore s = — (a + 6) ( 1 — cos — —rr ) ,
a ^ '^V a + 26/
measuring the arc and the deflection from the cusp A and
the tangent at the cusp.
16 ROULETTES.
Hence 8 : 2vers.arcP0 :: 2.00 : OP^
the form given in the Prmcipia, section x., prop. XLViii.
We may observe that the radius of curvature of the
epicycloid at Q
dif) a + 26 ^ a + 2b ^
22. To find the equation, in p and r, of an epicycloid.
If OQ = r, we have, from the triangle OCQ,
r» = (a + 6)» + 6"-26(a + 6)cosy
= a" + 46 (a 4- 6) sin" ^
and therefore o* = 4n rr (^ — ^*)«
-^ 46 (a + 6) ^ ^
23. ^Ac hypocycloid.
If a circle of radius 6 roll inside, or, more generally, with
its concavity in the same direction, on a fixed circle of
radius a, and if (fig. 14) Q is the tracing point, QE is the
tangent at Q to the hypocycloid.
Supposing Q originally at A, so that OA is the tangent
at the cusp A, let ^ be the deflection of the tangent at Q
from the tangent at A,
Then, if AOC = 0,
p = 07=(a- 26) sin CEQ = (o - 26) sin g ;
and jj = (a — 26) sin _jL, .
BOULETTES. 17
Siuce the curve is convex to the point 0,
da d*p 46 (a — 6) . a<b -.
d4 ^ d4* a-26 ^"^a-26 '
and 4&(a-&) / g^ \
If 6 > a, this may be written in the form
dit> *^^ ''^a + 2(6-a)^''a + 2(6-a)'
so that, when 6 > a, the hypocycloid is identical with the
epicycloid generated by a circle of radius 6 — a rolling
outside a circle of radius a.
This can also be seen by direct geometry; for if PQ
meet the fixed circle in R (fig. 15), let OR produced meet
DQ produced in E\ then RE is the diameter of a circle,
touching at R and passing through Q.
The angle FQR = FRQ = RPO; therefore FQ is parallel
to OP ; and
ZREQ = '^^ERQ = PDQ;
:. RE=0E''0R = 2{b- a),
and OF^h^ CQ, so that OF is parallel to CQ.
Hence arc JBQ = (6 - a) . POR = (6 - a) PCQ
= (6 — a) . -7- = arc RA,
so that the point Q, carried by the circle F, will produce the
hypocycloid.
24. It may be useful to give the several equations for
the case of a three-cusped hypocycloid, or tricusp, a curve
possessing many remarkable properties.
* The p and r equation is ¥^=^-r — rT(a"-'^)» and, if 26=a-c, this
becomes pl^ = -5 5 (a" - r% the form given by the Jesuit Fathers in the
notes to Prop. li. of the Principia.
B. B. 2
18 ROULETTES.
If AOP = 0, and if <f> is the angle of deflection from OA,
the tangent at -4, of EQ the tangent at Q (fig. 14),
QGP = S0,BJid(f> = PEQ^POA=^-e = ^,
Taking 3c as the radius of the fixed circle,
p = 0r=0^sin(7^Q = csin^,
.'. p = csin3<^,
is the tangential polar equation of the tricusp, referred to its
centre.
The curve being convex to the point 0,
ds
= —p — -j^ = 8c sin 3^,
d<t> ^ d<t>'
.'. 5 = f c (1 — cos 3<^),
is the intrinsic equation of the tricusp, measuring 8 and (f>
from a cusp and the tangent at the cusp.
TT
Writing^ H--^ for <^, and assuming that 8 and -^ vanish to-
gether, we obtain
5 = f c sin 3*^,
which is the intrinsic equation of the tricusp referred to the
middle point between two consecutive cusps, and the tangent
at that point.
Again, from the triangle OCQ, if OQ = r,
r* = 4c' + c' + 4c'cos3^ = 9c'-8c'sin»y,
.•.7^ = 9c*-8p»,
is the equation, in p and r, of the tricusp.
26. Properties of the tricusp,
(1) The portion of the tangent within the curve is of constant
length, and the locus of its middle point is the circle inscribed
in the curve.
ROULETTES. 19
Let il be a cusp of the tricusp described by the point P
(fig. 16).
If EQ is parallel to DP, the point Q is a point on the
tricusp.
For, since the inclination of DP to OA is 2^, it follows
that DP turns round twice as fast as OD, and in the
contrary direction, and therefore that Q is the position of P
when OD has turned through two right angles.
PQ is equal to DE and is therefore of constant length,
and if OBT is parallel to DP, B is the middle point of PQ,
and the locus of B is the circle centre and radius OB,
Now describe the circle, diameter BT, and draw TO
perpendicular to PQ.
Then aPB = 2(|-05P) = 7r-2(7r-3e) = 6^-7r.
.-. arc TBC = 6cd = arc TA, since AOB=^ 20.
*
.'. the point (7 is a point on the tricusp, and PQ is the
tangent at C.
(2) TTie distance between the centres of curvature corre-
sponding to the intersections of the tangent with the curve is
constant.
If K is the centre of curvature at P, in PR produced,
PZ = 8csin3</> = 8csiny = 4PP.
and, if K^ is the centre of curvature at Q, QK' = 4Q5.
Hence it follows that KK' is parallel to PQ, and that
ZZ' = PQ + 4fifi = 65i,
so that KK' is double the diameter of the fixed circle.
26. (3) The envelope of the pedal line of a tricvngle is a
triiyusp.
(Quarterly Journal of Pure and Applied Mathematics, No. 38, 1869.)
Taking any point P in the circumscribing circle, centre
2—2
20 ROULETTES.
0, let PKy PL be the perpendiculars on the sides AC, AB,
and NY the perpendicular from N, the middle point of
AC, on KL, which is called the ' pedal line/
Then if OJV^r= </>, and COP = 6 (fig. 17),
and NY =NK Bin (f>
= JBsin<^sin(£+2^ + 2<^-7r),if OP = jB,
= I [cos {^ - (0 - ^)} - cos {3^ - ((7- ^)}].
Now, if 0' be the centre of the nine-point circle,
ON(y = C^A,
and therefore, taking p as the perpendicular from 0' on iJ2',
P = -^cob{S<I>-(C^A)}
= — -^cos3<^',
changing the initial line.
This is the 'tangential polar' equation of a three-cusped
hypocycloid, generated by a circle of radius -^ rolling inside
M
a circle of radius-^ .
At
Hefrvce the envelope of the pedal line of any triangle is a
three-cusped hypocycloidy the centre of which is the centre of
the nine-point circle.
We may remark that if KP be produced to meet the
circle in p, the line Bp is parallel to KL ; a simple method
is thus found of constructing the various positions of KL.
The question considered is a particular case of the follow-
ing problem :
BOULETTES. 21
Perpendiculars PK, PL are let fall on two fixed straight
lines OJ., OB ; given the locus of P, it is required to find the
envelope of KL.
To do this, let jp, the perpendicular from on KL, make
with OA the angle <f>, and let POA = 0,AOB = a.
Then p = OK cos <^ = OP cos ^ cos ^ ;
and (f>=:LKP = LOP = a-0;
therefore if OP =/(^),
p =f(oL — <!>) COS (a — <^) cos ^
is the equation to the envelope.
The problem of the * pedal line' has been discussed in this
Journal, by Messrs. Greer, Walton, Ferrers, and GriflSth;
it was, I think, first pointed out by Mr. Ferrers that the
centre of the tricusp is the centre of the nine-point circle.
27. It may be noticed that the trilinear equation of the
tricusp, referred to the triangle formed by the three cusps, is
111/.
^/a Vp V7
that its tangential equation is
(u + v + wf = 27wvw;,
and that its reciprocal polar with regard to its centre is
rsin3^ = c.
Curves rolling on fixed curves.
28. To find the path of any given point in the area of a
plane curve which rolls on a fixed curve.
If O'P be the rolling curve, 0' having been coincident
with 0, fig. (18), let X, y, be coordinates of P referred to the
normal and tangent at 0, and x , y\ referred similarly to 0' ;
<f>, (f>', the angles which the normal at P makes with the
normals at 0, 0'.
22 ROULETTES*
Then, if a, /8 are the coordinates of 0', referred to 0,
a = y' sin (<^ + ^') — x' cos (<^ + ^') — a?.
^ = y-y'cos(<^ + <^')-a?'sin(<^ + <^0^
and, as the right-hand members of these equations can be
found in terms of the arc OF (5), the relation between a and
/S can be found.
If Q be a point, the coordinates of which, referred to 0',
are (a, 6), and if f, 17, be the coordinates of Q referred to 0,
f = a — a cos (<^ -f <^') "" ^ s^^ (<A + ^0
17 = /8 -f a sin (^ -f <^0 + ^ cos (<^ -f ^')-
29. If QP = r, and PQO' = ^, the relation between r and
Q is the polar equation of the rolling curve referred to Q and
(iO\
Taking f and 17 as the coordinates of Q referred to 0, and
-i^ as the angle Q-EiV,
frsrcos-^ — a?, and 97 = rsin'^ + y.
Now tan-f = -^, and tanPrO = ^,
.-. r^=tanQPr=tan(P^r-fP^'-iEO
dx d't)
1 + ^ . ^'
dx ' df}
If the fixed and rolling curves are given, j- can be formed in
terms of r, and -r- in terms of x, and then the elimination of
dx
X and r will give the differential equation of the locus of Q.
If the locus of Q and the fixed curve are given, the
elimination of x and ^ will give the rolling curve.
If the locus of Q and the rolling curve are given, the
elimination of x and f ^ will give the fixed curve.
ROULETTES. 23
30. A curve rolls on a straight line; it is required to
find the curvature of the path of any point carried with it
Let QP, Q[P be consecutive normals to the path of
Q (fig. 9).
Their intersection E is ultimately the centre of curva-
ture.
Let QP be the changed position of Qp,
Then Q'LQ = the angle through which the curve has
turned
= the angle through which the normal at P
has turned
= — , if p be the radius of curvature at P ;
P
. r.o_ arcQ(y _ QQ
••^'^"" Z.E '^QLQ'^LQP
P
OS Ss . cos a '
p r
if a be the angle between QP (r) and the normal at P,
r — pcosa'
ultimately, observing that since the displacement of p from
P' is of the second order, we may in this case assume that
p and P' are coincident ; Arts. (3) and (4).
Thus, in the case in which the roulette is a cycloid,
r = 2a cos a, and p = a;
.-. ^Q = 2r = 2PQ.
31. The focus-rovlettes of a conic section on a straight
line.
Let Q, fig. (9), be the focus of a conic rolling on a
straight line.
24 ROULETTES.
Then /:> cos a is the chord of curvature through the
focus.
If the conic is an ellipse,
CD" r(2a-r)
pcosa^—^—-—,
and therefore, if p' be the radius of curvature of the focus-
roulette,
111
-, + - = -.
p r a
If the conic is an hyperbola,
r (2a + r)
p cos a = — ^^ -.
and therefore -, + - = — ,
p r a
shewing that the roulette is convex to the straight line.
If the conic is a parabola,
p cos a = 2r,
and therefore p'=:^r,
the known property of a catenary, the directrix of which
is the straight line.
We can however deduce the equation of the catenary.
For, if <f> is the inclination of the tangent at Q to the
fixed line, ^ = a,
and — = 1 — cos 2 f ^ — <^] = 2 cos' (f).
Hence /M ~ ^ " ^ ®®^' ^'
and therefore 8 = a tan <^.
32. The preceding formula may also be obtained by
help of the equation
BOULETTTES. 25
For (fig. 19), if OPQ = <^, and OP = AP = s,
p^r — 8cos(f>,
and -~ = OZ =8sm(f>;
But if AQP = 0, and AFP = ^fr,
^ = ^ + |_<^;
therefore radius of curvature of roulette
J . du . , , cJ^
and since r -7- = sm 9, and -^j = p,
^, . . cos a r*
this =r-f
1 1 r — p cos a
cos a
p r
We may observe that the curvature of the roulette is
zero if r = p cos a, that is, if the point is situated on the
circle of which p is the diameter.
33. The following theorem is of great importance.
If a cv/rve roll on a fixed curve over a small arc &, the
angle turned through by any line in the plane of the rolling
curve is is {- -\- -A y where p and p are the radii of curvature
of the fiaed amd rolling curves.
Let Pp = PF = hs, fig. (20), and let the normals OP,
O'P, meet in i.
Then, since OP turns into the position KLP', the angle
OLK between the lines is the angle required, and this angle
^POp^FOP
__ tiS Ss
p p
26 ROULETTES.
If the concavities are in the same direction, and if the
rolling curve is inside the fixed curve the angle turned
through is
but, if the rolling curve is outside the fixed curve, the angle
turned through is
\p 9 1
34. A curve rolls on a fixed curve ; to find the curvature
of the roulette traced by any point carried with it
Q being the carried point, the angle QLQ = the angle of
displacement = & f - + -7), (fig. 21).
^^'QEQ-QLQ-PQL'
and, since the displacement of ^ is of the second order,
nriT PP" coaa Sscosa
PQL pQ —:^,
taking a as the inclination of QP to the normal at P.
r(- + - )
^ 1 1 cos a
-+
P P r
If the curve roll inside the fixed curve the expression
will be
\p pl
1 1 cos a *
P P ~r~
We observe that the curvature of the roulette vanishes if
r = pp cos a/ip + p')y
and therefore there is a point of inflection of the roulette if
ROULETTEa 27
the point P is situated on the circle of which pp'l{p 4- p) is
the diameter.
35. Referring to the same figure, let -^ be the inclination
of QP to the fixed normal OF, A the point which has passed
over 0,^QP = e,QP = r.
Then, if r = f(0) is known, and it s = J?'(^) is the equa-
tion of OP, EQ can be found in terms of 8 and therefore
of <^.
Also '^ = <^ — a ;
,\ -Tj or EQ can be found in terms of -^j and this gives the
intrinsic equation of the path of Q.
36. Thus, for an epicycloid,
radius of curvature of path of Q (fig. 12),
1 1
^PQ ^"^"^
1 1 cos a *
a'^b'"PQ
1 1
Again, take the case of an ellipse rolling on an equal
ellipse, corresponding points being in contact, and consider
the path of a focus 8.
If P be the point of contact,
PF , cjy.
9^ SP
•'• ^ = 27^7 TT" " ^^^'
CI/ AC.8P
as is d priori obvious.
28 ROULETTES.
37. If a cwrve roll on a straight line, the arc of the
roulette is eqtial to the corresponding arc of the pedal.
The angle turned through (fig. 22),
= angle between normals at P and jp,
= 7(27';
.-. arc Qq = QP.7QY'
= QT.TQ7'
YT' ,
" sin Y^Y' ' ^^ " ^^' '
QT being the diameter of the circle about YQY'.
Or thus, iiQY=y,
^ = cos QPY,
and if <7 = arc of pedal, y being the radius vector,
^ = COS QYZ= COB QPY;
CuO"
.'. ds^^da,
38. If a curve roll on a fixed curve, the element of arc
of the roulette is to the corresponding arc of the pedal as
p + p' : p\ p being the radius of curvature of the rolling curve,
and p' ofthefi^ed curve.
Imagining the line OT to be the fixed curve (fig. 22),
the angle turned through = cfo f - + - J ;
<P P
Also, QT being the diameter of the circle TQT,
Y7'=^QT.smYQT^QP.-;
P P P
BOULSTTES. 29
Hence the length of the arc of the roulette
where (Jo* is an element of the arc of the pedaL
In the case of cycloidal, or trochoidal curves, p and p'
are constant, and the arc of the roulette is proportional to
the arc of the pedal.
In the case of a curve rolling on an equal curve, corre-
sponding points being in contact, the arc of the roulette is
always double that of the pedal.
Also the roulette of any point is similar to, and double
of the pedal.
39. If a curve roll on a straight line, the area betiveen
the roulette, the fixed line, and any two ordinates, is dovble
the corresponding pedal area.
For, if F'JV be the perpendicular from F' on P F (fig.
22), the element of area = QY. YN, neglecting infinitesi-
mals of the second order,
= QF. FF'. cos TFF
^QY. FF'. sin QFr
= 2AQFF'.
Or thus, if X, y be co-ordinates of Q,
y~^y^mQPY^y^mQYZ^QZ^p\
,'. ydx — pds^pda-
= 2 (element of polar area).
40. To find the area swept over by the normal QP.
Taking figure (22), let QP = r, PQp = B0, and S<l> = the
angle of deflection from P to p, which is equal to the angle
between Qp and Q^P',
30 ROULETTES.
Then the area QPF Q' = QPP' + QF Q
= QPp + QpQ.
observing that pP" is of the second order,
therefore the area between the roulette, the fixed line and
two normals
If the curve be a closed curve and make one revolution,
1 [^
this area = area of curve + „ I ^'c?<^.
Hence, if the rolling commence when QP is perpen-
dicular to the fixed line,
1 [^
2 (area of pedal) = area of curve + ^ / r^d<f> ;
or
area of curve =1 (i^* "" 9 ^) ^^•
Take for example the case of a cycloid.
The area swept over by QP, fig. (4),
= 3a» [cos* ^-d0 = ^{0 + sin 6),
and, if 5 = TT, this = ^ 7ra*,
so that the whole area of the cycloid is 37^a^
41. A curve rolls on a fixed curve ; to find the area swept
over by the normal QP.
ROULETTES. 31
If the arc Pp^PF (fig. 23), then, pF being of the
second order,
area QPP'(2' = QPp + QpQ'
therefore area swept over by QP
Hence, if the curve be a closed oval, and if it make a
complete revolution, the area between the arc of the roulette,
the two normals at its ends, and the curve
= area of curve + ^ p S* f- + ~) •
42. To find the locus of the centre of curvature at the point
of contact of a curve rolling on a straight line.
Let X, y be co-ordinates of the centre of curvature, then,
if 8 =f((l>) be the rolling curve,
x = s=f((l>\
and y = P=f (</>),
whence, eliminating <f>, the locus is obtained.
Thus, if the curve be an epicycloid, or hypocycloid, the
locus is an ellipse.
If it be a catenary, the locus is a parabola, and if it be an
equiangular spiral, the locus is a straight line.
43. If the curve roll on a fixed curve, s = -P(<^'), and
if s =f{<t>) be the rolling curve,
^=OJf+/t>sin<^'(fig. 24),
y = pcos<f>' — PM ;
therefore, if OM and PM can be found in terms of <^', we
have, with/(<^) = -P(<^'), three equations from which ^ and
<^' may be eliminated.
32 ROnLETT£S.
Suppose, for example, the curves to be equal catenaries,
their vertices at first coinciding.
Then, /> = c8ec'<^, and PJIf =csec0 — c;
The locus is therefore a straight line, as is d priori
obvious, if we remember that the normal bounded by the
directrix is equal to the radius of curvature.
If the curves be equal cycloids, their vertices at first
coinciding,
OP = 4a sin <^, PE = 4ia cos <^,
OJf = a (2<^ 4- sin 2<^), and PJf = a (1 - cos 2</)).
.-. OiV=2a<^ + 3asin2<^, and N'E^a-}-Sacos2<l>,
so that the locus of E is the same as the locus of a point
carried by a circle of radius a rolling on a straight line, the
point being at the distance 3a from the centre of the ciltle.
44. To find the length of the curve formed by the
successive positions of the centre of curvature we may
proceed as follows.
Let Q be the centre of curvature at P, q at p, and Q' the
position of q when the curve has rolled from P to P', so that
QQ' is an element of the locus (fig. 25).
Then q may be taken to be in the normal PQ, since its
distance from PQ, the tangent to the evolute at g, is an
infinitesimal of the second order.
Hence, if PQ = p, and if p be the radius of curvature of
the fixed curve at P,
r
and the inclination to PQ of the tangent at Q
pdp
ROULETTES. 33
Ako, if QQ = S<r,
Hence we can find the intrinsic equation, for if yft be the
inclination of the tangent QQ' to the tangent at a fixed
point 0,
<l> being the deflection of the fixed curve from to P.
As an example^ again take the case of the equal cate-
naries; then
p^p'^c aec*<f>y & = pS<l),
and -^^ = <^ - - + tan"^ (cot <^) = 0,
so that QQf is parallel to the tangent at 0, as already seen.
45. Envelope Rcmlettes,
We have hitherto considered only the roulettes produced
by points carried with a rolling curve ; we now proceed to
consider the roulettes enveloped by lines carried with a
rolling curve.
A curve rolls on a straight line, to find the envelope of any
straight line carried with it
If P be the point of contact, and PQ the perpendicular
let fall from P on the carried line, Q is the point of contact
of its envelope (fig. 26).
Let pq be the perpendicular from a consecutive point »,
then as the curve rolls over PP', q is carried to Q', and if o<r
be an arc of the roulette enveloped,
Ba = Qq + qQf
= sin <f>Ss + rS<l),
if OPQ = </>, and PQ = r.
B. B. 3
34 KOXTLETTES.
Hence the radius of curvature = t-,
d<l>
, ds
= r + /> sin 0.
For example, consider the roulette produced by a diameter
of a circle rolling on a straight line.
Then r = a sin <^, p = a,
and the roulette is a cycloid.
Ex. 2. A parabola rolls on a straight line, it is required
to find the envelope of the lotus rectum.
In this case, p sin ^ = 28P (fig. 27), and
PQ = a — a? ;
.'. 'r:=a'-x + 2(a-\-x)
= 3a + a cot'<^,
.*. = 2a + a cosec*^ ;
.'. <r = 2a^ — a cot ^ + C,
and the length of the roulette between the two points at
which it cuts the fixed line, i.e. from
9=T to <p = — , IS (7r + 2)a.
46. The tangential polar equation may be obtained,
directly.
Thus, if « = OP = arc -4P, and p = OY the perpendicular
on the carried line,
p = r — s cos <f>,
whence the equation, if r and s be known in terms of <^.
ROULETTES. 35
Or, if 8 only be known, and OZ be the perpendicular
on PQ, fig. 26,
and therefore ^ = 8&m<f>.
Hence the radius of curvature
= r — 8 cos <l> + 8 cos <p + sin -^
= r + p sin 0,
as before.
47. A curve rolh on a given curve, carrying a 8traight
line; to find the roulette enveloped.
Making the same construction as before, and observing
that the displacement of _p is of the second order (fig. 28),
= 88 COS a + rB<f>,
where S0 = &(-+ -,], PQ = r and a is the inclination of
PQ to the normal at P.
Hence the radius of curvature of the roulette at Q
da- , pp'
a<t> P"^ P
and if r, a, p and p' can be found in terms of the angle which
PQ makes with some fixed line, the intrinsic equation can
be found.
If the concavities are in the same direction, the expres-
sion for the radius of curvature of the envelope roulette is
PP
r + cos a -P^—
P -P
3—2
36 ROULETTES.
48. Example. A circle rolls outside a fixed circle; to
find the length of the curve enveloped by a diameter.
If AD be the diameter, A passing over A\ and if
AVP = (fig. 29),
PQ = 6sin^,
J da , . aO , r,-Dr\ ^
and j^ = 6 sin -J- + cos CFQ — — ,
a<p a + o
6' + 2a6 . ad
a + o
The angle of deflection of AD from OA*
= ^ = ^Tf-y;
da V + 2a6 . a6 , .
d<f> a + b a + b ^ ^
and <r: -_ (^1 - cos ^^ j .
measuring from A',
Taking a half roll of the circle, that is from ^ = to
= — , we get the length of the arc from one cusp to the
a
next, which is therefore
2 - (6 + 2a).
a
Comparing the equation (a) with that of Art. (21), we .
observe that the envelope of AD is the epicycloid which I
would be produced by a circle of radius ^ rolling on the
circle 0.
This can also be seen geometrically, for, describing a
circle on PC as diameter, the arc PQ is equal to the arc pA,
and therefore to the arc PA\
ROULETTES. 37
49. A curve rolls on an equal curve, corresponding points
coinciding; to find the envelope of any normal of the rolling
curve.
Let = BTQ (fig. 80), and measure -^ from the tangent
at 0, the point corresponding to A.
Then, for the envelope of the normal at -4,
-^ = r + cos a| , Art. (47),
and, if $ =f(y^) be the curve,
r = j cos y^ds, p =/' (^), and <^ = 2-^ ;
J
•■•l-zl-^/'w+^-t/'d)-
For example, let a circle roll on an equal circle ; then
da- [2 ^T^ a . (b 3a , d>
and or = 3a ( 1 — cos ^
a two-cusped epicycloid.
Taking a half roll of the circle the arc is 6a, which
agrees with the result in Art. (48), putting 6 = a.
50. A curve rolls on a fi^xed curve; it is required to find
the envelope of a/ny curve carried with it.
If P be the point of contact, draw from P normals to
the carried curve (fig. 31).
Then, if PQ be one of these normals, it is the normal
at Q to the envelope, and the other normals similarly belong
to other portions of the envelope.
38 ROULETTES.
Let pqy the normal from a consecutive point p, roll into
the position P'Q \ then E, the intersection of PQ and P'Q',
is the centre of curvature.
The arc QQ^Zcr^ Qq + rS<}>, where r^PQ and
Let p'' be the radius of curvature at Q of the carried
curve; then
Qq : S^ . cos a :: />" : r + p",
a being the inclination of QP to the normal at P;
and the angle PEP" = 80 - ^
therefore EQ, the radius of curvature of the envelope,
p COS a
-e%')
l^?
cos a
r + p"
Making />" infinite, this of course gives the formula of Art.
(47), and if />" vanish, the formula of Art. (34) results.
If the various quantities involved in this expression can
be found in terms of y^^ the angle of deflection of PQ, the
intrinsic equation is determined.
If any of the curves instead of being convex, as in the
figure, be concave, the signs of p, &c. must be changed.
51. Ex. 1. A curve rolls on another, carrying a paraMel
curve.
In this case, a = 0, r = rf, and p" ^p-^ d.
Hence EQ becomes (2 + />', a result which is obviously
true.
BOULETTES. 39
Ex. 2. A circle, of radius c, rolls inside an oval curve.
If Scr be an elementary arc of the envelope, and & the
arc of the curve rolled over,
&7 = 2cSif> - Bs,
where S^ = S« f j ,
so that Sot = & — 2c — ,
P
and therefore total length of envelope = — 47rc, where p is
the perimeter of the oval curve.
Ex. 3. A straight line rolls on a fixed cirde, carrying an
equal circle with which it is in contact.
Let A'OP = (fig. 32), then
r^CP-a^aJlT0^-a,
a 1
cos a = T^v^ =
cp jr^'
//
and therefore t-t- = a
p^co, p =p =a,
d^r - 1
1-
1 + ^
=n — ^ — J-
Also, if •^ be the inclination of PQ to OA',
.-. d-<jr = d0-Y:^,
This equation, when integrated, determines the length of
the envelope.
40 ROULETTES.
If we put —a for p\ and OF'\-a for r, we shall obtain
the other portion of the envelope due to the normal PQ'.
52. A curve rolls on a straight line; to find the area
between the straight line, the envelope of any carried straight
line, and two normals of the envelope.
The element of area PQ(^P' (fig. 26),
^PQqp + FqQ'
= rBs sin <^ + ^ r^B(f>
= (rp sin <^ + 2 rM S^,
the integration of which expression gives the area if the
intrinsic equation of the rolling curve be known.
If the line PQ fell below the line OP, the element of
area swept over will be
Irp sin <^ — ^ r* j S<^.
This however is included in the former, if we suppose
r to be an algebraic expression for PQ.
53. A curve rolls on another; to find the area between
the envelope of any carried curve, any two normals of the
envelope, and thefiaed curve.
The element of area = PQQ'P' (fig. 31),
= PQqp + lr^^,
where S</) = Ssf- + — j.
Hence the area swept over exceeds the area between the
curve and the carried curve by / 5 i^d<f>.
ROULETTES. 41
Thus in the case of Ex. 3 of Art. 51,
and dyjt^d0-Y:^y
therefore the area of the envelope exceeds the area APQ by
jj(2 + 0'-'2jl + 0')dylr.
54. Adopting the notation of preceding articles, we can
give an expression for the element PQqp (fig. 31).
For this element
and &" : & . cos a :: p" : r + p'^ ;
therefore element
r\r cos a&
PQqp = {p'' + i)
r + p
55. The following examples will serve as additional
illustrations of the preceding methods.
A cycloid rolls on a straight line; it is proposed to consider
the roulette enveloped by the tangent at its vertex.
The cycloid is s = 4asin</), and for the envelope o{ AQ
(fig. 33),
^ = pcos <^-PQ
= 4a cos* ^ — 2a sin' <}>
= a + 3a cos 2^ ;
*. <r = a9 + -5- sm 20,
measuring from 0.
42 BOULETTES.
To trace the curve, observe that there is a cusp when
cos 2<^ = - g ,
ie., when <^ is a little greater than q, and that the curve
cuts the initial line at distances 2a from : also that for one
roll of the cycloid the curve lies wholly below the initial line.
If the cycloid be continued, and the rolling go on con-
tinuously, the next branch of the cycloid will give the next
half of the roulette above the initial line, after which the
successive branches of the cycloid will produce continually
the same roulette.
Further, the evolute of the roulette is
5 = 3a cos 20,
a four-cusped hjrpocycloid.
The curve may be further examined by finding x and y,
the co-ordinates of Q referred to 0, viz.
^ = 4a sin ^ — 2a sin" <f>, y = a sin ^ sin 20.
Fig. 34 represents the roulette, and its evolute. ,
The element of area swept over by PQ
= PQS5cos0-|pQ*S0
= 4a" sin* (2 - 3 sin' <f>) S<f>,
2
which becomes negative when sin' > ^ , or when
cos 20 = - g ,
i.e. at the cusp.
2
The integration from = to sin' = « gives the area
2 TT
OET, and from sin'0 = o ^^0 = 2 the difference between
the areas AET and GET.
BOULETT£S. 43
If we wish to find the area enclosed by the roulette, it
will be at once given by integrating the expression^
^{xdy — ydx), from ^ = to <^ = k.
This expression
= c? (6 sin* <^ — 4 sin* ^) d<^,
the integral of which between the assigned limits
= a^6.j.2-4.2J2=8«'
and the complete area inclosed by the roulette =-7ra*, i.e.
it is half the area of the generating circle of the cycloid.
To find the length of the arc of the roulette, we must
take ^ from to sin"*A/^, and then from = 8in"*A/^ to
0= ~, and add together the numerical values of the resulta
This will give one-fourth of the whole arc.
If <l> be taken from to 5- we obtain <r = -^ , so that the
difference between the arcs OE and EC of the roulette is one
quarter of the perimeter of the generating circle of the cycloid.
56. A circle rolls on an equal circle, carrying a tangent;
it is required to determine the nature of the roulette produced
by the tangent
Let OY-=^p (fig. 35), A'OP = ACP j=0,a,nd 70A' ==<}>,
Y being the perpendicular on the carried line.
Then ^ = 2acosd — a, and <^ = 25;
.'. p = 2a cos ^ — a,
the tangential polar equation of the envelope.
44 ROULETTES.
T^T ^ da <Pp 3a <f>
.'. <r = 3a sin ? — a0,
the intrinsic equation, measuring a from A'
Observing that the radius of curvature
3a ^
= -^ cos a — a,
2
we see that a cusp occurs when cos ^ = ^ . There are therefore
two cusps, corresponding to the positive and negative values
o{0.
When d = |, p = 0.
When cos5 = ^, it will be seen by a figure that the
tangent passes through the other end of the diameter through
P, and that the envelope then crosses the circle at a point V
2a .
distant -^ from P. It will also be found that the tangent at
the cusp meets the diameter A'O in the same point T at
which it is intersected by the envelope itself.
Putting together all these considerations we obtain the
figure (fig. 36), the curve being an involute of a two-cusped
epicycloid
The element of area swept over by PQy i.e.
PQQ'P' = r cos aSs - 1 r'Stf)
= rhO {a cos a — r)
= rS^ (2a cos ^ - a)
=prS0.
When 0>^ , this expression is still the same if we write
o
for p its numerical value a — 2a cos 0, and any portion of the
area is thus found by a simple integration.
EXAMPLES. 45
EXAMPLES.
1. If a cycloid roll on the tangent at the vertex, the
locus of the centre of curvature at the point of contact is a
semicircle whose radius is four times that of the generating
circle.
2. Prove that a cardioid is an epicycloid due to the
rolling of a circle, with internal contact on a fixed circle of
half its diameter.
3. The roulette, on a straight line, of the pole of an
epicycloid is an ellipse.
4. The roulette, on a circle, of the pole of an equiangular
spiral, is an involute of another circle.
5. When a curve rolls on a straight line, shew how to
find the locus of the centre of curvature at the point of
contact, and prove that, in the case of a cardioid, the locus is
an ellipse.
When a curve rolls on a fixed curve, prove that the locus
of the centre of curvature of the rolling curve at the point of
contact is inclined to the common tangent at the angle
tan"* \pdpl(p'\- p)d8]y where p, p are the radii of curvature
of the fixed and rolling curves at the point of contact.
6. A curve A rolls on a curve B so that its pole describes
a straight base, and the curvatures of A and B at the point
of contact are always as n to 1, estimated in the same
direction. Prove that the radius of curvature of jB is w — 1
times the normal terminated by the base, and that the chord
of curvature of A through the pole is to the radius vector
in the ratio of 2 (n — 1) to n.
Prove also that if A and B roll on a straight line, the
roulette of the pole of A is the envelope of the base carried
by fi, and that the radius of curvature of the roulette is n
times the normal terminated by the same straight line.
State what curves A and B are when w is, — 1, 0, 1, 2, 3.
46 EXAMPLES.
7. A cycloid rolls on an equal cycloid, corresponding
points being in contact ; prove that the locus of the centre
of curvature of the rolling curve at the point of Qontact is a
trochoid whose generating circle is equal to that of either
cycloid.
8. A circle rolls on a straight line; prove that the
envelope of any carried straight line is an involute of a
cycloid ; and trace the figures corresponding to the cases
in which the distance of the carried line from the centre
is greater than, equal to, or less than the diameter of the
circle.
9. A straight line rolls on the curve, s =/' (<^), carrying
a straight line inclined to it at the angle a ; the envelope
roulette is
s = sin af (<^) + cos a/' (<^).
If the curve be an epicycloid, or hypocycloid, the envelope
roulette is of the same class.
10. The roulette, on a straight line, of the pole of the
hyperbolic spiral, rO = c, is
ft
dy^ y .
dx Je-f'
and of the pole of the curve, <fp = r**', is
11. The roulette, on a straight line, of the pole of a
cardioid is
4a - a; = {2 (2a)* + y*} {(2a)* - y*}*.
15
and its area is -^ ira\
12. A parabola rolls symmetrically on an equal para-
bola ; find the path of the focus, and prove that the path of
the vertex is the cissoid
EXAMPLES. 47
13. An involute of a circle rolls on a straight line ; the
roulette of the centre of the circle is a parabola.
14. The ellipse - = 1 + e cos ^ rolls on a straight line ;
the path of the focus is given by the equation
, - 2c da? . c*
y ds f'
and the path of the centre by
2a and 26 being the axes.
15. The roulette, on a straight line, of the centre of a
rectangular hyperbola is
16. A cycloid rolls on a straight line ; the locus of its
vertex is given by the equations
07 = 2a (sin ^ — <^ cos <^), y = 2a(f> sin <^,
the origin being the point of the line over which the vertex
passes.
17. A curve rolls symmetrically on an equal curve,
carrying an involute ; the envelope of this involute is an
involute of the fixed curve.
18. If a curve roll on a straight line, the curvature of
a point roulette varies as -j- {-), p ai^d r being referred
to the point.
If the curve be
- = 1 + sec a sin (0 sin a),
T
the roulette is a circle.
48 EXAMPLES.
19. The curve P'Q rolls on the curve PQ, P' passing
over P ; the roulette of P' is, in the neighbourhood of P,
a semi-cubical parabola, of which the parameter is
^PP (p + p)
p and /}' being the radii of curvature at the point of contact.
20. A catenary, s = c tan <^, rolls symmetrically on an
equal catenary ; the intrinsic equation to the envelope of its
axis is
da- , , ir — ylrc. yfr ylr
j-i- = c log tan — T-*- +s tan ^ sec -^ .
dyjr ^4222
21. If an oval curve roll on a straight line, prove that
the area traced out by any point in the curve will exceed
1 r2ir
the area of the curve by ^ I r^d<f>, where r is the distance
from of any point P of the curve, and <^ the angle which
the tangent at P makes with some fixed line in the curve :
apply this to find the area of a cycloid.
22. If an oval curve A roll upon an equal and similar
curve B, so that the point of contact is a centre of similitude
for each, then the whole area traced out by any point when
A has made a complete revolution, is twice the area which
would have been traced out if the curve A had rolled upon
a straight line.
23. Test the formula of Art. (40) by applying it to
an ellipse, measuring r from the focus.
24. A parabola rolls on a straight line from one end
of the latus rectum to the other; the length of the arc
enveloped by the axis is
2a log (26).
25. A parabola rolls symmetrically on an equal parabola,
from one end of the latus rectum {4a) to the other ; the length
of arc enveloped by the axis is
2a log (4e).
EXAMPLES. 49
26. A diameter of a circle rolls on a curve ; the envelope
of the carried circle consists of two involutes of the curve.
27. A circle, radius b, rolls on a fixed circle of radius a ;
the area between the fixed circle, and the envelope of a
diameter for a half roll from one end of the diameter to the
other is equal to
g(3a + 6).
28. The lemniscate r^ = a* cos 26 rolls on a straight line ;
the tangential polar equation of the roulette produced by its
pole is
-^ +JP tan <^ = a tan <^ Vsin ^ ;
and the intrinsic equation to the envelope of its axis is
jx = i 1 5 sm 6 - 3 sm -g .
/
cos
29. A circle rolls on a fixed circle ; the envelope of any
carried straight line is an involute of an epicycloid.
30. A catenary rolls on a straight line ; the envelope of
any carried straight line is an involute of a parabola.
31. An ellipse rolls, symmetrically, on an equal ellipse ;
prove that the whole length of the arc enveloped by its
axis is
(^l-f— ^log
2a . -. . .^^ -
1— ^y
62. A curve, carrying a point, rolls on a straight line,
and then, symmetrically, on an equal curve ; prove that after
rolling over the same arc in each case, the radii of curvature
of the roulettes,„and the distance of the point from the point
of contact, are in Harmonic Progression.
33. In the same case, if a straight line be carried, the
radii of curvature of the roulettes, and the distance of the line
from the point of contact, are in Arithmetic Progression.
B. R. 4
50 EXAMPLES.
34. If a given arc of a curve roll, first externally, and
then internally, over the same arc of a fixed curve, the sum
or difference of the arcs of the roulettes of the same point
is independent of the nature of the fixed curve; the sum,
when the radius of curvature of the fixed curve, at each point
of contact, is greater than that of the moving curve, and the
difference, when the reverse is the case.
The same independence also exists for the sum, or
difference, of the areas swept over by the straight line joining
the carried point with the point of contact.
35. A parabola, latus rectum 4a, rolls on a circle of radius
6, the rolling commencing at the verte'x. Prove that when
the parabola rolls to the end of the latus rectum the corre-
sponding arc of the roulette enveloped by the axis is
a (1 + log 2) + 3^ (2^2 - 1).
36. An ellipse rolls on a straight line ; the length of the
envelope of its axis between two consecutive cusps is
^('+^'^1^3-
37. Find the envelope roulette of the directrix of an
ellipse which rolls on a straight line ; and prove that it has
two cusps if the^ eccentricity is greater than ^ (>/5 — 1), and
that, if e < i (Jo — 1), the length of the arc of the roulette,
corresponding to a complete roll of the ellipse, is equal to the
perimeter of a circle, the radius of which is equal to the
distance between the directrices of the ellipse.
38. The envelope roulette, on a straight line, of the
axis of a rectangular hjrperbola, is given by the intrinsic
equation
a , n/2 + 1 acos6
« + a = -f= log -,= -. -f
J 2 J 2 cos <f> + s/cos 2<^ ^/cos20
GLISSETTES.
57. Glissettes are the curves traced out by points, or
enveloped by curves, carried by a curve, which is made to
slide between given points or given curves.
Thus if an ellipse slide between two fixed straight lines
at right angles to each other, the glissette traced by its
centre is the arc of a circle.
Again, if a straight line, of given length, slide between
two fixed straight lines at right angles to each other, the
glissette of any point in the line is an ellipse.
In this case, if p be the perpendicular from the inter-
section of the fixed lines on the sliding line (length 2a), and
<f> its inclination to one of the fixed lines,
jp = a sin 2<^ ;
the envelope-glissette is therefore a four-cusped hypocycloid.
58. A curve slides between two straight lines at right
angles ; to find the glissette of any carried point
Let the tangential polar equation of the curve, referred to
the carried point, be
then, if x, y be the perpendiculars from the carried point on
the two fixed tangents,
a?=/(<^),andy=/(^<^H-|),
and the elimination of ^ will give the rectangular equation of
the glissette.
4—2
52 GLISSETTES.
If the two fixed lines be not at right angles, but inclined
to each other at an angle tt — a, and if Xy y be the oblique
co-ordinates of the carried point, we shall have to eliminate
between the equations
X sin a =/(^), y sin a =/(<^ + a).
Ex. An ellipse slides between two straight lines inclined
to each other at an angle tt — a ; to find the path of the
centre.
In this case, we can write the equations in the form
a?' sin* a = a' cos' (0 "" « ) + ^* sin' (^ ~ 9) '
if sin' a = a' cos" (<^ + 1) + &* sin' ( <^ + 1) >
and the result of the elimination is
{(aj»+y') sin' a - a' - 6'}'+(a;' - 3^')' sin' a cos' a=(a' - 6')'cos' a,
59. The following theorem is of great importance.
Any state of motion of a plane area in its own plane
can be represented by a state of rotation about a point
Any plane area is fixed if two of its points are fixed; and,
if the motions of two of its points are given, the motion of
the area is given.
We must observe, however, that we cannot assign an
arbitrary motion to two points ; the restriction existing that
the velocities of the two points in the direction of the line-
joining them must be the same.
Suppose that two points P and Q are in motion in the
directions PT and QF, and that PE, QE are drawn perpen-
dicular to those lines respectively, and meeting in E,
It is clear that the motion of P may be represented by a
state of rotation about any point in the line PE, and that of
Q by a rotation about any point in QE.
Hence, both motions are represented by a state of rotation
about E.
OLISSETTES. 53
TbJB point E is called the instantaneous centre of rotation,
and the motion of any point It in the area is, at the instant
considered, at right angles to EE,
Thus, if a curve slide between two given lines, the
intersection of the normals at the points of contact is the
instantaneous centre.
60. Any motion of an area in its omn plane can be
represented by the roUing of a certain determinate curve on
another determinate curve.
Let PQRS (fig. 37) be the curve traced out in space
by the successive positions of the instantaneous centre, and
let the angular velocities of the moving area corresponding to
each position of the instantaneous centre be known.
Then, if P, Q, R, ...he successive positions of the centre
at given infinitesimal intervals of time, the lines in the
moving area QP, RQP\ SRq'p\ ... will turn successively into
the positions QP', Rqp\ Srqp, &c.
Hence the motion can be represented by rolling the curve
pqrs ... on the curve PQRS
But the curve pqrs... ia the locus on the moving area
of the instantaneous centre, and the two curves are therefore
determinate.
These curves are sometimes called the fixed and moving
centrodes.
61. Ex. 1. A straight line AB, of given length, slides
between two fijxed straight lines at right angles to each other.
In this case the locus of the instantaneous centre, Ey with
regard to the fixed lines OX, OY, is the circle, centre and
radius OE, which is equal to AB, fig. 38, while the locus of
E with regard to AB is the circle on AB as diameter.
The motion is therefore the same as that produced by the
rolling of a circle, internally, on a fixed circle of double its
radius.
The eflfect of this motion is discussed in Art. (7), and
54 OLISSETTES.
therefore it follows that the path of any point Q, connected
with AB, is an ellipse of which is the centre.
It has been shetvn in Art. (57), that the envelope of the
sliding line is afour-cusped hypocycloid.
We can prove this result also by direct geometry.
For if QCQ\ fig. 39, is the sliding Kne, and if EP is the
perpendicular fi'om E upon QCQ', it follows that P is the
point of contact of the envelope of QQf, and therefore that the
locus of P is the envelope.
It is clear that P is a point on the circle whose diameter
is CE, and CE is half of OE, and is therefore one-fourth of
the diameter of the fixed centrode EA,
Further the arc EP of the small circle is equal to the arc
EQ of the rolling centrode, and therefore equal to the arc
EA of the fixed centrode.
Hence it follows that the path of P is the path of a
point in the perimeter of a circle rolling inside a circle of four
times its radius, and is therefore a four-cusped hypocycloid.
Further we can see that the envelope of any straight line
through (7, making a given angle with QQ' and carried with
it, will be a four-cusped hypocycloid.
For the ends of this new diameter of the rolling centrode
will move along fixed radii of the fixed centrode, and the new
diameter will be under the same conditions as QQ\
If any other straight line, not passing throvgh C be carried
by QQ' the envelope-glissette will be an involute of a four-
crisped hypocycloid.
Let the line be at the distance c from C, and inclined at
the angle a to QQ\ and let jp be the perpendicular from upon
the line and the inclination of the perpendicular to OA.
Then 2a being the length of the sliding line
p = c + a sin {2(f> + a),
.•. the radius of curvature of the envelope
= c — 3a sin (2(f> + a).
6LISSETTES. 55
80 that the envelope is an involute of a four-cusped hypo-
cycloid.
62. The same results are true whatever be the angle
between the two fixed lines.
For if this ande is a, OE being the diameter of the
circumscribing circle of QEQ is equal to 2a cosec a, and is
therefore constant.
The fixed centrode is therefore the circle, centre and
radius 2a cosec a, also since QEQ is constant, the rolling
centrode is the circle QOQ^E, the radius of which is a cosec a.
63. Ex. 2. Two straight lines, AB, AC, containing a
given angle, move so that AB, AC pass respectively through
fixed points P and Q.
The point E being the instantaneous centre, AE is the
diameter of the circle about PAQ, and is constant. Fig. 40.
Hence, the moving centrode is the circle, centre A and
radius AE, while the fixed centrode is the circle PAQ.
The sliding motion is therefore equivalent to the motion
produced by rolling a circle, with internal contact, on a circle
of half its radius.
•
The point-glissettes are therefore hypotrochoids, and the
line-glissettes are circles.
This last can be shewn by the formula of Art. (14), for if
EN be the carried line, at the distance CNfrom C, OPQ = a,
p' = a, and /> = — 2a, fig. 41 ;
.-. ^ = PQ+ 2a cos OCA' = CN,
dip
Le. the curvature is constant.
Or, by direct geometry we can see that if A roll from A\
and Q be the other end of the diameter A' 00,
GQ = CN,
and therefore the envelope of EN is a circle, centre and
radius CN.
56 QLISSETTE&
If the straight lines AB, AG slide on fixed circles, the
results are exactly the same.
For the straight lines drawn through the centres of the
circles parallel to AB and AC meet at the same angle, and
the sliding of these lines through the centres carries with it
the motion of BAG.
64. Ex. 3. An involute of a circle slides between two
straight lines at right angles to each other.
Let Ox, Oy be the fixed lines, fig. 42, and let x and y be
co-ordinates of E, the instantaneous centre.
Then a;=(^+0ja — a, and y = a<^ 4- a ;
so that y — aj = f2 — ^Ja.
In this case the fixed centrode is a straight line, and the
moving centrode is a circle, centre C.
The point-glissettes are therefore cycloids and trochoids,
and the line-glissettes are evolutes of a cycloid.
. 65. A curve, carrying a point Q, slides on a fixed curve,
the same point P of the moving curve being always in contact
with the fixed curve.
In this case the instantaneous centre is the centre of
curvature E at P, and therefore QE is the normal to the
Q-glissette.
If the curve slides over the arc PP* of the fixed curve,
and Q moves over the elemental arc QQ, and if E' is the
centre of curvature at P', Q*E is the consecutive normal, and
F is the centre of curvature at Q. Fig. 43.
If PEF = i^, PE ^p,PQ = c, and if a is the inclination
of PQ to the normal at P,
QFQ=B<I> p =80-8/)-^^.
GLISSETTES. 57
Let p* = radius of curvature at Q,
and /}" = radius of curvature of the evolute at E\ then
\ _qFq; qFq
__ 1 (2p c sin a ^
1 1 p" c sin a
i.e.
9 (p« + c" + 2c/3 cos a)* (/>• + c' + 2cp cos a)*
In this case the fixed centrode is the evolute of the fixed
curve and the moving centrode is the normal at P, so that
the motion is equivalent to the rolling of a straight line on
the evolute.
If the sliding curve be a straight line, and the point Q a
point in the sliding line, the angle a is a right angle, and the
curvature of Q is given by the equation
tt
9 c
9 {p^^ef (p'+c'f
66. To find the point-glissettes and the line-glissettes, when
a curve slides so as to touch a fi^ed straight line at a fi^xed point.
P being the carried point, the relation OP =^f(PY) is
known, and therefore if OY = x and P Y= y, fig. 44,
is the point-glissette.
Again, if AP is the carried line, and OZ the perpendicular
upon it from 0, let
0^ = jp and ^Or='f.
Then, if TAP = a, and s =/(<^) is the intrinsic equation
of the curve referred to A and the tangent at A
and jp = I cfe sin = I /' ((f>) sin 0ci<^.
58 OLISSE'ITES.
From these equations we can obtain p in terms of yjr, t.e. the
tangential polar equation of the line-glissette.
In this case the instantaneous centre E is the centre of
curvature at 0, so that the fixed controde is the straight line
OE, and the moving centrode is the evolute of the curve.
67. A given point A. of a straight line moves along a fixed
line Ox, and the straight line passes through a fi^xed point C,
at the distance CO, (c),froni Ox.
The instantaneous centre, E, is the point of intersection
of the straight line through 0, perpendicular to AC, and of
the straight line through A perpendicular to Ox.
Taking a?, y as the co-ordinates of E referred to Ox and OC,
we observe that if 6 is the inclination of CA to 0-4,
a? = c cot 0, and y ^c + xcot0,
so that the equation of the fixed centrode is
^ = c (c - y).
Again if x, j/ are the co-ordinates of E referred to AC and
the perpendicular through A to AC,
c = x' sin 0, and x' = y' tan 0,
so that the moving centrode is the curve
and the motion is equivalent to the rolling of this curve on
the parabola, a? = c*^ cy»
68. A straight line passes through a fijxed point O, and a
given point A of the line moves along a given fijxed curve.
Taking for the origin, let r=f{0) (fig. 45), be the
equation of the given curve.
Then, E being the instantaneous centre if OE = r',
and EOy = ff, r', ff are co-ordinates of a point of the fixed
centrode.
Now d'=d,andr =rtan<^ = ^=/(^),
:. the fixed centrode is r ^f{ff).
OLISSETTES. 59*
Again, ii AE=p,
p = r sec ^ = sec <l>f(0),
^ . Idr f(e)
and the moving centrode will be obtained by the elimination
oi0.
69. A given point A of a straight line moves dUmg a fixed
line Ox, and the moving straight line always touches a given
fixed curve.
The point E being the instantaneous centre, let AE=^f>
and EAP = ^, fig. 46, then a?, y being the co-ordinates of P,
the point of contact,
/) = ilPsec^ = ysec'<^=y|l+(^)|,
and cot ^ = — -^ .
Hence eliminating x and y, we obtain the polar equation
of the moving centrode.
For the fixed centrode,
^ = OA=x + ytQ,n<f>=:x-y ^,
'=-^=»^(l)]'
and the elimination of x and y will give the Cartesian
Equation of the fixed centrode.
70, HolditcKs Theorem.
If a straight line APB, of given length a + b, moves with its
ends A and B on the arc of a closed oval curve, and if AP = a>
and BP = b, the difference between the area of the oval, and the
area of the locus of the point P, is equal to irah.
Let Q be the point of intersection of AB with a consecu-
tive position inclined to AB at the angle B0.
Then if CT be the area of the oval, V the area of the locus
€0 OLISSETTES.
of P, and TT the area of the locus of Q, that is, of the envelope
of AB, the diflference U — Wis equal to 5 1 AQ^dO, and is
also equal to = I (a + J — -4Q)" dft
Hence, if -4 Q = r,
j 'rd0 = I /^(a + 6) d^ = TT (a + 6) ,
also F- F=||p(yd^ = |f (a-r)»d^,
••. 1^-^=1/ {^-(«-^)'}^^ = |/(2a^-«")
rf^
= ttTT (a + 6) — Tra* = irab.
This proof was given in the Quarterly Jmi-ml of
Mathematics, vol. 2, 1858.
71. Amaler's Planimeter.
Amsler's Planimeter is an instrument employed to deter-
mine practically the area within any closed curve on a plane
surface, as, for instance, the area of an estate marked out on
an ordnance map.
It consists of two straight rods, AB, BC, jointed at B, and
capable of free motion about the end A, which is fixed, with
a small wheel, the plane of which is perpendicular to BG,
capable of turning round BG as an axis.
The centre of the wheel is sometimes at the end B of
BG, and sometimes at some point P between B and G, but
it might be placed at any point of BG produced either way.
The important point to be observed is that the plane of
the wheel is perpendicular to BG, so that the revolution of
the wheel determines the motion of the point B in the
direction perpendicular to BG.
We will first take the case in which the fixed point A
is outside of the contour line enclosing the area to be
measured.
GLISSETT£S. 61
Let AB = a, BG=^b, and let ij> and ^ be the inclinations
of AB and BG to some fixed direction in the plane of motion.
Fig. (47).
When the wheel, radius c, is at B, let the end G pass
over an elemental arc GG' of the contour line, so that ABG
takes up the position ARG'.
If dO is the angle turned through by the wheel, the area
ABGG'RA^^a'd<l> + ^b'dyfr + b.cd0;
for, neglecting infinitesimals of the second order, it is made
up of the triangle ABB', the parallelogram BGDR, RD being
parallel and equal to BG, and the triangle BDG\ Fig. (48).
When G has moved completely round the contour line,
and has returned to its initial position, the complete area
swept over by ABG will be the area enclosed by the contour
line, and this
= [(i a'd4> + ^ Vd^ + hcdO)
= hcO = hs,
where s is the total length passed over by the point B in the
direction always perpendicular to BG,
When the centre of the wheel is at P, let dO^ be the
angle turned through by the wheel while AB turns through
d(f>, BG moving parallel to itself into the position B^D, and
diS^ the angle turned through while B'D turns through d'>^
into the position BG\
Then cdO^ is the distance between the parallel lines BG,
BDy and cdd^ is the distance QP' perpendicular to BfQ,
Therefore the area ABGG'RA
= ^a'd<f> + ^Vdylr'hbcd0^
= I a'd(l> + (I Vdylr - bcde^ + be {d0, + d^J.
62 GLISSBTTES.
But cde^ = QP' = idf , if BP = I,
,-. the area ABCG'EA
if ds is the element of the distance passed ovfer by P in the
direction perpendicular to EG,
Hence, as before, the area enclosed by the contour line
= 65,
where 8 is the total length traversed by P in the direction
always perpendicular to EG,
We will now consider the case in which the point A is
inside the contour line.
Adopting the same notation as in the previous case.
The area enclosed by the contour line, fig. (49),
=|j| a^d<f> + he (d0, + d0,) + (i 6*^ - bij dyjr I
If the point B moves outside the closed curve, as in
fig. (50), the area enclosed is the difference between the
areas swept over by AB and BG; i.e. the area
=j^a*d<f,-lb'(-d^)-b(-cdd,)Y
Observing that 0^ and 0^ are now registered negative,
and that BG is also turning in the negative direction, we
have the relation,
ci-dd,)=i(-dn
.: the area =1(2 «W + 5 ^*d^ + bcdBA
= [jl a'd<l> + ^ Vd^lt - bcdd^ + be (dd, + ddA
= Jjl a*d<l> + (1 6' - W) df + bdal,
OLISSETTES. 63
ds being the element of the distance traversed by P in the
direction perpendicular to BO.
Hence as before the expression for the area is
72. A given curve slides between two straight lines at
right angles to eaoh other; to find the locuSy tvitii regard to the
lines, of the instantaneous centre.
Let X, y be co-ordinates of E, C the centre of curvature
at P, and CK an element of the evolute at P (fig. 61).
Then, by turning the curve round E through a small
angle S(f>y and taking K such that the angle of deflection of
the arc GK is 8^, the tangent at K will glide into the position
K'P", and we have
&p = -PP' = -(p-y)S<^;
dx
Similarly, we shaU'obtain
if p' be the radius of curvature at Q.
Hence, if the intrinsic equation be known,
p=f{4>) and p' =/ (<^ + 1) .
and the two equations we have obtained will give x and y
in terms of ^, and reduce the solution of the problem to
the elimination of (f>.
Take, for instance the case of Art. (64), in which
64 QLISSETTES.
and therefore Ti^y — " ^4>y
cup
Integrating these equations, and remembering that when
^ = 0, ^ = -ft — ct, ai^d y = a,
we shall obtain the equations of Art. (64).
73. The preceding equations determine the tangent and
normal at E, and, by differentiation, the curvature of the
locus of E.
A geometrical construction may however be given, for
since EN=PF ^EGh<f>, and E'N^EG'Bif) (fig. 52),
/. EN : E'N :: EG : EC,
and the angle NEE = ECU.
Hence, if EF be perpendicular to EE,
the angle FEC = | - E'EN = ECF,
and F is the middle point of GC
If then D, D' be the centres of curvature consecutive
to (7, C in their new positions, and F' the middle point
of DUy the lines EF, EF' intersect in the centre of curva-
ture at ^.
For the carried locus, i. e. the locus of E with regard
to the sliding curve, we observe that the tangent and normal
are the same as for the locus above considered.
For the curvature, however, let GK, CK\ fig. 53, be
elements of the evolute having the same deflection 8^, E' the
intersection of the tangents at K and K\ and F' the middle
point of KK' ; then Q, the intersection of EF and E'F\ is
the centre of curvature.
GLISSETTES. 65
In either case, CK being an infinitesimal of the first order,
arc EE' = EF. 2B<f> = CC'B<f>,
or ultimately -j-. = GG\
Also, if the FECT in (figure 52) ='^,
, EC
tan^ = ^^,
and, if EC and EC, i. e. p — y and p — x, can be found
in terms of <}>, we shall have sufl&cient data for the deter-
mination of j^ > ^1^® radius of curvature.
(Up
For the carried locus (fig. 53), we must add to -^ the
inclination of EC to some fixed line in the moving area.
74. If a right angle slide on the arc of a (ywrve, p = f (^),
the motion is equivalent to that of rolling the curve
upon the curve
of = cos 4>f' (^ + 1) + sin ^/' (^),
For (fig. 54) if P and Q be the points of contact, the in-
tersection E, of the normal is the instantaneous centre, and if
TP = x, TQ=y, X and y are the coordinates of the carried
locus of E, and since
07 =p^f {<!>),
Vr=/(^+|)aadQr=/(^+|).
Hence the first two equations follow at once.
B. R. 5
66 GLISSETTES.
For the fixed locus of E let fall the perpendicular EN on
the initial line from which <f> is measured; then if ON=x'
and EN=y'y the second pair of equations is obtained.
As an example, let the given curve be a three-cusped
hypocycloid, so that
^ = a cos 3^.
Then it will be found that
^ + 2/'=16a* and a?" + y'' = 9a^
so that the motion is produced by rolling a circle of radius
4a, with internal contact, on a circle of radius 3a.
This result can be obtained from direct Geometry.
For (fig. 55) if F, G, be two positions of the centre of the
rolling circle on a diameter QOQ\ the tangents TP, TP' at
the points P and P' of the curve will be at right angles;
and the normals QP, Q'P' will meet at a point E on the
circumference of the fixed circle. It will be easily seen that
the line TE passes through 0, and is four times the radius
(c) of the rolling circle, so that while the fixed locus of E is
the circle of radius 3c, the carried locus is a circle of radius
40, and having T for its centre.
It will be seen that the equations of Art. (72), i.e.
dx dy , ,
follow at once from the preceding equations.
75. Taking the general case of any motion of a plane
area (which however is reducible to a case of roulettes), and
supposing that the centrodes can be found, or, which is the
same thing, the fixed centrode, and the rates of rotation as
compared with the increase of arc along the centrode, we
can find simple expressions for the curvatures of the roulettes.
To find the curvature of the roulette of a point Q.
Let EE {ha) be an element of the fixed centrode fig. 56,
and a the inclination of EQ (r) to the normal at E.
GLISSETTES. 67
Then VQ : EQ :: QQ' : QQ^EN,
if EN is perpendicular to EQ ;
or radius of curvature = VQ
rd^ — do- . cos a *
If we take the case of Art. (34),
and we obtain the formula of that article.
If we take the case of Art. (73),
da = CO'8^,
and .'. radius of curvature =
r - cos a . CC '
76* To find the curvature of an envelope-roulette.
Let r = EQ be the perpendicular from E on the line
fig. 67.
Then & = Q? + yQ' = S<j-cosa + ^QS<^,
1 ds da
and jT = ^ + cos a j7 .
a9 a<p
For instance in the case of Art. 73, the radius of curvature
= r + cos a . GC
77. To find the curvature of the envelope of any carried
curve.
Let EQ, Eq be normals to the carried curve, and p" the
radius of curvature at Q, fig. 58.
Then & = Q? + Qq
-^<^<^os a ^,^ + EQS<l>,
and if (2FQ' = 8^,
8<^-SVr = ^0^=^"^'"
p' + r '
5—2
68 GLISSETTES.
therefore, radius of curvature of envelope
_ da
dyjt
p" cos a , , , .
^-77 da- + rd6
^ p +r 3.
,. cos ado-
For instance, taking the case of Art. (60), we have
d<^ = da(- + —] ,
and we fall upon the formula of that article.
78. A triangle moves in a plane so that two of its sides
slide on fixed curves ; it is required to find the envelope of
the third side.*
Let a, )S, 7 be the perpendiculars from any fixed point on
the sides a,bfC; then
aa + 6)S + C7 = 2 (area of triangle) = 2A.
Hence, if ^ be the angle which one of the perpendiculars
makes with any fixed line, and remembering that
(fa
is an expression for the radius of curvature of the envelope
of the side BC, we obtain
ap^ + 6/Oj + cpg = 2A,
where p^, p^, p^ are the radii of curvature of the envelopes of
the several sides.
If then two of these be given, the third is determined.
79. Ex. 1. A given triangle moves so that two of its
sides touch fiaed circles.
In this case p^ and p^ are constant ; /ql is therefore con-
stant, and the third side always touches a fixed circle.
* The method of this article is due to Dr Ferrers.
QUSSETTES. 69
This includes the example of Art. (63) as a particular
case.
A direct geometrical proof may be also given*.
Firstly, let the sides AB, AC pass through fixed points
Through A draw AF parallel to BG, and meeting in F
the fixed circle which is the locus of A, fig. 59,
The angle FAP = ABG, a^d therefore the arc FP is
constant, and i^ is a fixed point.
Also, the perpendicular FG from F on BC is equal to
the altitude of the triangle ABC; therefore BC always
touches a circle, the centre of which is at F.
Secondly, let AB, AC touch fixed circles having their
centres at P' and Q\
Through P' and Q' draw lines P'A\ QA' parallel to AB,
AG, and meeting BG in B' and G' : then, as before, B'C'
touches a fixed circle.
80. Ex. 2. A triangle tnoves so that two of its sides
slide on the arc of a fixed cycloid.
In this case
Pj = d sin ^, and p^ = d sin (^ + a) ;
.'. cpg = 2A — od sin ^ — 6d sin (^ + a),
which can be written in the form
P3 = 6+/sin(<^ + /3).
Hence it appears that the envelope of the third side is an
involute of a cycloid.
81. Two straight lines AB, AC, inclined to each other at
a given angle X, and carrying a line AD, slide on fixed curves;
it is required to find the envelope of AD.
This is a particular case of Art. (78), and if fi, v be the
* Several proofs were given, by myself and others, in the Educati(mal
Times for 1864.
70 GLISSETTES.
inclinations of AD to AB and AC, the first equation of that
article becomes
a sin/Lt + )S sin i; + 7 sin X = 0.
Hence p^ sin fi + p^smv + p, sin X = 0,
and if p^ and /o, are given, p^ is determined.
82. Ex. 1. Thus, from Ex. (2), Art. (79), we obtain
the following result.
If two straight lines at right angles to each other slide
on the arc of a cycloid, the straight line bisecting the angle
between them always touches a cycloid.
Ex. 2. AB, AC slide completely round an ovcd curve.
In this case, if
. ^ do" . ds . cfe'
da, ds' being elements at the points of contact of AB, AC.
Hence, if ? be the perimeter of the oval, and a- the whole
arc enveloped by AD,
sin X . <r = i (sin /a + sin v),
. fl + V
sin -2-
or o- = 6—
Sm ^-rr —
We have put
da
d<l> ^«'
because if a and /8 are positive, 7 is necessarily negative.
EXAMPLES. 71
EXAMPLES.
1. P is a fixed point on the circumference of a given
circle, and PQ any chord drawn through P is produced to R,
so that QR is of constant length. If PE be drawn per-
pendicular to QP to cut the circle again in E, and RE be
joined, shew that RE is normal to the locus of R.
2. Prove that a lima9on, r = a + 6cos^, is a point
roulette produced by the rolling of a circle, with internal
contact, on a fixed circle of half its radius.
3. The angle BAG slides over two fixed circles; prove
that the point-gUssettes are lima5ons.
4. A parabola slides on two straight lines at right angles
to each other; prove that its vertex and focus respectively
describe the curves,
^y'(^ + y' + 3a") = a', and a?y = a* (a;* + y*).
5. C is a fixed point external to a given circle whose
centre is 0. TEB is a tangent of given length, touching
the circle in E\ its extremity T being in CO produced. If
CB and OE when produced intersect in if, shew that the
length of BG is least when TM is at right angles to TG.
6. A plane moves in any (given) manner on a fixed
plane : is a fixed point on the fixed plane, P a fixed point
on the moving plane ; if the area described by P about
is given, shew that the locus of all points (P) in the moving
plane for which this area is the same is a circle, and that for
diflferent values of the area the corresponding circles are
concentric.
7. A given triangle moves in a plane so that one of its
sides toucnes a fixed circle, and another a fixed cycloid ;
prove that the third side touches an involute of a cycloid.
72 EXAMPLES.
8. Two tangents OPy OQ inclined at a constant angle a
are drawn to a closed curve, without points of inflexion, and
OX is the external bisector of the angle POQ. Shew that
the perimeter of the closed curve enveloped by OX : the
perimeter of the original curve : : 1 : sin ^ot.
9. One end ^ of a straight line slides along a fixed
straight line OA, and the straight line always passes through
a fixed point G at the distance CO, (c), from the line OA ;
prove that this motion is equivalent to the rolling of the
curve, c* (i»* + y^) = a?*, upon the parabola, cy^a^ + c*.
10. If a parabola, latus rectum 4a, slide between two
stmght lines at right angles to one another, the glissettes
produced are the same as the roulettes produced by a
parabola latus rectum a, rolling on the curve
11. A straight line slides on a curve having always the
same point in contact ; the motion is identical with that
of rolling a perpendicular straight line on the evolute.
Hence shew that the envelope of the straight lines drawn
through each point of an epicycloid at a constant angle to the
tangent is also an epicycloid.
12. An ellipse slides on a straight line, always touching
it at the same point ; the path of its centre is the curve
a?y' = ia'-y')(f-b').
13. A straight rod, of length 2a, slides with its ends on
a wire in the form of the cardioid, r = a (1 — cos tf) ; prove
that the fixed and moving centrodes are circles.
14. A straight line ACB slides on a fixed curve, the
middle point G being always in contact with the curve ; if
1/p is the curvature at (7, and 1/r and 1/s are the curvatures
of the paths of A and B, and if AB = 2c, prove that
EXAMPLES. 73
15. A given point in a straight line moves along a given
diameter, produced, of a given circle, and the straight line
always touches the circle ; prove that this motion is equivalent
to the rolling of a parabola upon the curve,
ay = (a? + a)' (2cw? + a?),
the parabola starting with its vertex at the origin.
16. The end P of a finite line PQ travels on a closed
curve which has no points of inflexion, and the line PQ is
inclined at a constant angle to the normal at P. Find
the area included between the locus of Q and the curve,
and shew that if be the centre of curvature at P the
motion of Q is perpendicular to QO.
17. Two tangents, inclined to each other at a given
angle, move round a closed curve without cusps, or points
of ii^exion, and the external angle is divided into two
constant parts a and /8; prove that the length of the
envelope of the dividing line is to the length of the given
curve, as
cos -^ IS to COS — ^ .
18. One end of a straight rod moves round the cir-
cumference of a circle, and the rod always passes through
a fixed point of the circumference ; prove that this motion
can be produced by rolling a circle with internal contact
upon a circle of half its radius.
19. If a straight rod pass through the vertex of a
parabola, and one end move along the arc of the curve,
shew that this motion can be produced by rolling the
curve a"y* = (a?* + 4a') {a? + 4a* {a? + y^} upon the curve
Say' = a;* (a? — 4a).
20. An involute of a circle slides on a straight line,
always touching it at the same point ; the glissettes of a
point and a straight line are respectively a trochoid and
an involute of a cycloid.
74 EXAMPLES.
21. A parabola slides on a straight line touching it
at a fixed point P.
If the normal at P meet the axis in and OR be
drawn parallel to 8P and equal to one fourth of the latus
rectum, the normal to the path of the focus is parallel to
PR.
Shew also that the path of the focus is an hyperbola.
22. If at any point of a curve whose intrinsic equation
is 8 =/' (0) a straight line is drawn making a constant angle
a with the tangent and of length s cos a, the intrinsic equa-
tion of the locus of its extremity will be
8 =/' (0) sin ci±/(<f>) cos Of.
23. A fixed point of a straight line moves along the
axis, produced, of a parabola, and the straight line always
touches the parabola; prove that the motion is equivalent
to the rolling of the curve,
upon the curve, ay* = 4a? (a? + of.
24. A lamina moves in its own plane so that a point
fixed in it lies on a straight line fixed in the plane, and
that a straight line fixed in it always passes through a
point fixed in the plane; the distances from each point to
each line being equal. Prove that the fixed and moving
centrodes are parabolas.
25. A given right-angled triangle PQR is made to slide
round the outside of a fixed oval cui've with the point P on
the curve, the side PR touching it, and the side PQ normal
to it. If 5 is the perimeter of the oval, prove that the length
of the curve enveloped by QR is equal to
(« + 27r . PQ) sin PQR.
GENERAL MOTION OF A RIGID BODY.
83. A rigid body is said to have a motion of translationy
when all planes in the body move parallel to themselves ; or,
which comes to the same thing, when all points of the body
pass over equal distances in the same direction.
A rigid body is said to have a motion of rotation, or to
have rotatory motion, when some plane in the body changes
its angle of inclination to some plane fixed in space.
In general, the motion of a rigid body can always be
represented by a rotation combined with a translation, and
the translation may be rectilinear or curvilinear.
84. The wheel of a carriage, for instance, has a recti-
linear motion of translation, combined with a rotatory motion
about its centre ; but this, as we have seen, can be repre-
sented by a state of rotation about the point of contact of
the wheel with the ground.
If a man looking straight at a particular wall of a room,
walks round a table in the room, he has a motion of circular
translation, but no motion of rotation ; all the points of his
body moving in equal circles with different centres.
As another illustration, the moon moves round the earth
so as always to present very nearly the same face to the
earth.
It follows therefore that, while the centre of the moon
moves round the earth in its oval orbit, an ellipse of small
eccentricity, the moon turns completely round an axis
76 MOTION ABOUT
through its centre, that is to say, it has an angular velocity
of four right angles per month*.
The motion of the moon is therefore represented by a
motion of elliptic translation, combined with a motion of
rotation.
Again, when a mass of liquid rotates uniformly, as if
rigid, about a vertical axis, every molecule describes a circle,
and has besides a rotatory motion, the free surface being a
paraboloid ; whereas, in Rankine*s free circular vortex, every
molecule describes a circle, but has no rotatory motion, and
the free surface is convex, and has a horizontal asymptotic
plane.
Motion of a rigid body ahovt a fixed point.
85. If two straight lines OPy OQ, through the fixed
point 0, are fixed, it is clear that the body is incapable of
motion, and the motion of the body is therefore completely
determined by the motions of these two straight lines.
At any instant the point P is in motion in some definite
direction, and the line OP has a motion in the plane
containing this direction.
Drawing the plane POG through OP perpendicular to
its plane of motion, the motion of OP can be represented by
a state of rotation about any line through in this perpen-
dicular plane.
Similarly the motion of OQ can be represented by a state
of rotation about an axis in a plane Q0(7, intersecting the
other plane in the line OC. Both motions are represented
by a single rotation about the line OG,
The motion of a rigid body about a fixed point is there-
fore, at any instant, a motion of rotation about some axis
through the point.
* About thirty years ago a curious controversy took place in the colomns
of the * Times', concerning the motion of the Moon. It was asserted that,
because the Moon always presents the same face to the Earth, it has no
rotatory motion, and there was a good deal of correspondence before the
matter was settled. The misconception of course was in the use and meaning
of the word rotation.
A FIXED POINT. 77
86. The successive positions of the instantaneous axis
in space will form a cone, fixed in space ; and the successive
positions of the instantaneous axis in the body will form a
cone, fixed in the body, and the whole motion will be
represented by the rolling of this cone upon the fixed cone.
These are sometimes called the fixed and moving axodes.
An important application of this idea is the discussion of
the motion of a body under the action of no force, in
Poinsot's Nouvelle TMorie de la rotation des corps solides.
87. It may be useful to indicate a ditferent method of
dealing with the ideas of the two preceding articles.
Consider the body to be rigidly attached to a sphere, the
centre of which is at the fixed point ; then the motion of the
sphere will determine the motion of the body.
If P and Q are two points on the surface of the sphere,
the motions of P and Q determine the motion of the sphere.
Through P and Q draw great circles perpendicular
respectively, to the directions of motion of P and Q ; these
great circles intersect in a point C, which has no motion, and
OC is therefore the instantaneous axis.
And, exactly as in Art. (60), the locus of C on the surface
of the moving sphere will be a spherical curve rolling on the
arc of a fixed spherical curve, the locus of C in space, thus
constituting the fixed and moving spherical centrodes.
88. A circle rolls on the arc of a fixed circle^ the plane of
the rolling circle being inclined at a given angle to the plane
of the fixed circle ; it is required to fimd the position of the
instantaneous aans.
Take a and c as the radii of the fixed and rolling circles,
and a as the inclination of their planes to each other.
and G being the centres of the circles, the normals
to their planes through and G meet in a fixed point E,
fig. 60, and the motion is completely represented by the
rolling of the right circular cone vertex E, and vertical
angle PEQ, upon the fixed right circular cone, vertex E
and vertical angle double the angle OEP.
78 MOTION ABOUT
The line EP is therefore the instantaneous axis.
The angle OPQ being a, let the angle OPE = <f> ; then
angle
PEG=l-ia-<l>) = l-a + il>.
Projecting EC and OOP upon the plane of the fixed
circle,
EC ain a = a + c cos (tt — a) = a - c cos a.
.'. EP sin (a — <^) sin a = a — c cos a,
and a sin (a — <^) sin a = (a — c cos a) cos <^,
1 J- X ^ , c — acosa
leading to tan q> = ; .
® ^ a sin a
TT
If a is greater than ^ , <^ is positive, but if a is less than
^ , and A cos a > o, <^ is negative, and the instantaneous axis
PE is beneath the plane of the fixed circle, results which are
at once obvious from the figure.
The semi-vertical angles of the fixed and rolling cones
. TT TT
are respectively ^ — <^, and ^ — (a — <^), and the tangents of
these angles are respectively
a sin a •, c sin a
, and
c — a cos a a — c cos a
Again, if V is the velocity with which the point of
contact moves round,
the velocity of = F —
a
Therefore the angular velocity of the disc about the
instantaneous axis EP
-rrd — c cos a . . , V
= V r c sm (a — <p)
TT fl 1 2 cos a
[or c oc
i
A FIXED POINT. 79
In the general case of any plane curve rolling on another
plane curve, this is the relation between the angular velocity,
and the velocity of the point of contact, if a and c are the
radii of curvature at the point of contact.
89. The rate of rotation is the angular velocity about
the axis, or the spin, as it was called by the late Professor
Clifford. It is measured by the rate of increase of the
inclination of a fixed plane in the body, containing the
axis, to a fixed plane in space, containing the axis.
If OA is the axis of a spin, it is considered to be positive
when the motion is clockwise, looking in the direction AO,
or counter-clockwise, looking in the direction OA.
Composition of Rotations,
90. Parallelogram of angular velocities, or parallelogram
of spins.
If OA, OB are the axes of two spins, and if the lengths
of OA and OB are proportional to the magnitudes of the
spins, the resultant spin is represented in magnitude and
direction by the diagonal 00 of the parallelogram formed
by OA and OB.
Taking a> and to as the magnitude of the spins, the
velocity of any point P in the plane AOB, perpendicular
to that plane,
^wPD-^wPE, fig. 61,
PD and PE being the perpendiculars on OA and OB.
Now, if PD meets BG, or BG produced, in K,
OA.PD^OA{PK-\'KD) = BG.PK-¥OA.KD
^2(PBG'hOBG),
and 0B.PE^2P0B.
.-. 0A.PD'\-0B.PE^2.P0C=^0C.PF,
if PF is the perpendicular on OG.
80 RESULTANT OF SPINS.
Hence, if the ratio of Xi to © is the same as that of OG to
0-4, the velocity of P
and therefore OC represents the resultant spin in direction
and magnitude.
91. Residtant of two spins about parallel axes.
Suppose a rigid body to have, at any instant, two spins,
to and (o\ about parallel axes through the points A and A
Take any point P in the plane containing these axes,
and let PAK be perpendicular to the axes. The velocity of
P will be in the direction perpendicular to the plane, and its
magnitude will be
(oAP + (o'BP.
This will vanish if P coincide with a point C between A
and B, such that
a>AG=a>'BG,
and the velocity of P will be (© + m) GP,
It follows that the resultant spin is of magnitude © + ©'
about the axis through G pietrallel to the original axes.
If the spins about A and B are in contrary directions,
and of numerical magnitude o) and ©', the velocity of D
will be
oy.AP-ay'.BP,
and this will vanish if P coincide with a point G in BA
produced, such that
a}.AG==(D'.BG,
and the velocity of P will then be (to — m) GP.
If in this case a) = a)', the velocity of any point in the
plane will be
(o.AB,
and therefore, equal and opposite spins about parallel axes
are equivalent to, and may be represented by, a motion of
translation.
ROLLING SPHERE. 81
92. A sphere rolls on a plane so that its centre moves in
a circle.
Let P be the point of contact, the centre of the circle,
radius a, on which it moves, and C the centre of the sphere,
of radius c.
Then, if the sphere have no spin about the vertical
diameter, OP is the instantaneous axis, and therefore the
motion is represented by the roUing of the cone, axis OC and
semi-vertical angle GOP, upon the plane.
The cone is the moving axode, and the plane is the fixed
axode.
If Xi is the angular velocity of the point C about the
normal to the plane through the point 0, and if SI' is the
spin about the instantaneous axis, looking in the direction
PO, it follows that fi' is negative, and that its numerical
value is aSl/c.
93. If the sphere have a constant spin © about the
vertical diameter, a state of things which exists under certain
dynamical conditions, the instantaneous axis is the line EP,
such that
OJT: OP ::a>: ft', fig. (62),
and that the fixed axode is the right circular cone, axis EO,
and vertical angle OEP, and that the moving axode is the
circular cone, axis EG and vertical angle PEG, so that the
small circle PQ is the circle of contact.
94. A sphere rolls on a surface of revolution, so that its
centre moves in a circle.
If P is the point of contact, and if PO, the tangent to
the meridian at the point P, meets the axis Oz of the
surface in the point 0, PO will be the instantaneous axis,
provided that the sphere has no angular velocity about the
normal at P, a state of things which is possible.
In this case the fixed axode will be the cone, axis Oz,
and vertical angle twice POz, and the moving axode will be
the cone, having its vertex at the point 0, and enveloping
the sphere.
B. R. 6
82 SCREWS.
If the sphere should have a constant velocity about the
normal at P, which is dynamically possible, the instantaneous
axis will be a straight line through P meeting Oz in a fixed
point E, The fixed axode will then be the cone, axis Ez^
and vertical angle twice PEz.SiXid the moving axode will be
the cone, having EG for its axis, and CEP for its semi-vertical
angle.
Motions of Translation and Rotation combined.
95. If one point of a rigid body is fixed, it is clear that
the body cannot have a motion of translation, but that it
may have 2^ spin about some axis through the point
It follows therefore that any state of motion of a rigid
body can be represented by a motion of translation, combined
with a spin about some axis.
It will be shewn in the next article that this state of
motion can always be transformed into a translation in some
direction, combined with a spin about an axis in that direc-
tion, that is by a spin on a screw.
This screw is called the instantaneous screw.
The successive positions of its axis, in space and in the
body, create two ruled surfaces, which are called respectively
the fixed and moving axodes.
These axodes may be developable surfaces, or skew
surfaces, or, according to Professor Cayley's nomenclature,
torses or scroUs, and the motion of the body is completely
represented by the rolling and sliding of the one axode on
the other.
96. If we have given the state of the motion of one
point of a body at any instant, and the rotation of the body
about some axis through the point, it is clear that the motion
of the body is completely determined.
With these data we can find the instantaneous screw.
Let u be the velocity of a point of a body, and OA its
direction of motion, and let the body have the spin oi about
the axis OB.
SCREWS. 83
Draw the straight line OE perpendicular to the plane
A OB, fig. (63), and through the point E of the line draw EF
parallel to OB,
Apply to the body two equal and opposite spins, of
magnitude o), about the axis EF,
The spins a> and — » about OB and EF are equivalent to
the translation ©r, if OE = r, in the direction perpendicular
to the plane BOE,
The resultant translation will be in the direction EF, if
rto = u sin a,
a being the angle AOB, and if v is the magnitude of the
motion of translation,
V =^u cos a.
EF is therefore the axis of the instantaneous screw, and the
motion is represented by the velocity v in the direction EF,
and the spin to about EF,
97. It will now be seen that when the motion is repre-
sented by a spin on a screw, the magnitude of the translation
is the least possible.
In fact, if we apply two equal and opposite spins, each
equal to the screw spin, about any axis parallel to the axis of
the screw, we shall obtain a spin about this parallel axis and
a translation perpendicular to it, which, combined with the
original translation, will produce a translation of greater
magnitude.
98. Other modes of representing the motion of a body
may be adopted.
For instance, any state of motion can be represented by
two spins about axes at right angles to each other, in an
infinite number of ways. To prove this, take any point E
in the axis of the instantaneous screw EF, fig. (64), and, in
any plane through EF, draw two straight lines EA, EB at
right angles to each other, and draw PEQ perpendicular to
the plane AEB.
6—2
I
84 SCREWS.
K V and cd represent the screw motion, and if the angle
FEA = 0, these are equivalent to the two screws,
V cos 0, o) cos 0, and v sin 0, a> sin 0.
Through P and Q draw straight lines FO, QD parallel to
EA and EB,
Apply two equal and opposite spins, a> cos 0, to PC, and
two equal and opposite spins, co sin 0, to Qi).
If we take P and Q such that
QE . ft> sin ^ = V cos ^,
and PE, © cos ^ = v sin 0,
all translations will disappear, and we shall be left with two
spins, o) cos about PC, and o) sin ^ about QD.
Conversely if we are given two spins, a and /8, about axes
at right angles, and at a given distance c from each other, we
can determme the instantaneous screw.
For ft) cos ^ = a, and © sin ^ = /8 ;
.'. a) = ^/a* + /8^ and tan^= -.
a
Also, c:=PE+QE^ ^
ft> sin ^ cos
ca/8
v = c<»sin ^cos^= -, ,
and PE:QE:\^:aL\
so that the screw is completely determined.
Pitch of a Screw.
99. In the case of a screw motion, when v is the
velocity of translation, and © the spin, if we take a quantity
p such that
v^poa,
p is called the pitch of the screw.
If the motion continues uniform, p is the actual translation
due to a twist through the unit of angular measure.
SCREW& 85
Composition of Screws.
100. The axes of two screws intersect at right angles ; it
is required to find their resultant
Let v, CO and v', «' represent the two screws ; then if p
and p' are the pitches,
t;=pea and v'=jpV.
Ox and Oy being the axes of the screws, fig. (65), let OP
be the axis of the resultant XI of the two spins cd and o)'.
Also let OQ be the direction of the resultant V of the
two velocities v and v\
The motion is then reduced to the translation V in OQ
and the spin fl about OP.
Let the angle POx = 0, and QOx = <^.
In the line Oz, perpendicular to the plane xOy, take a
point E at the distance z from 0, and draw EF parallel to
OP,
Apply to the body two equal and opposite spins about
EFy each equal to fl.
The motion then consists of the spin fl about EF, the
translation Slz perpendicular to the plane EOP, and the
translation V in OQ.
The resulting translation is in the direction EF, if
zn = Fsin (<^ — 5) = v' cos ^ — -y sin 0,
leading to «fi' = (jp' — p)©©' (1),
or z={p' —p)sia0 cos 0.
This determines the position of the axis of the resultant
screw, the spin of which is fl, and the translation U, where
ft = J'^T^'\
86 CYLINDROID.
and i7= Fcos {(f> — ff) = vcos ^ + v'sin
= po) cos + p'oj' sin ^
= ft(|)cos'^+;>'sin'^),
so that if «r is the pitclj of the resultant screw,
vr=p cos* 0+p' sin* 0.
If we measure from the line bisecting the angle between
the axes of the screws, that is, if we write ^ + ^ for ^ we
obtain
2z = (p' — p) cos 20 J
and 2J7= n {/ +p + (/-jp)sin 2^}.
2%« GyUndroid,
101. If p and /)' are given, and if a> and to are allowed to
change, the angle will be variable, and the position of the
instantaneous axis will change.
It will trace out a skew surface, conoidal, of which Oz is
the axis, and, since the equations defining its position are
z = {p —p) sin ^ cos ^, y = x tan 0,
it follows by the elimination of that
^(^'^y^)=(p'-p)^
is the equation of the conoidal surface.
This surface is called a cylindroid, in the nomenclature
of Sir R. S. Ball (Ball's Theory of Screws).
Turning the axes of x and y through half a right angle,
the equation takes the form
102. The axes of two screws intersect at the angle 2a ;
it is required to find their resultant.
Take Ox and Oy bisecting the angles 2a and ir — 2a, the
C7LINDR0ID. 87
screws p and p' having for their axes OA and OB respec-
tively.
The translations are (t/ + 1;) cos a, and (v'— i;)sina, and
the spins about Ox and Oy are
(a + co) cos a, and (oi' — an) sin a.
Taking il and F as the resultants and 0, <^, as the inclina-
tions to Ox, we have
F' = t;» -h v'" + 2w cos 2a, ft" = ©" -f ©'* + 2a>a>' cos 2a.
Fcos <^ = (v' + v) cos a, i2 cos ^ = (©' + cd) cos a,
Fsin ^ = (v' — v) sin a, fi sin ^ = (©' -- cd) sin a.
As in Art. 100, if we take zD. = Fsin {(f> - 0),
we obtain ^ft' = (|j'— jp)(»tt>'sin 2a (1),
which becomes equation (1) of Art. 100, when AOB is a
right angle.
The spin of the resultant screw is Xi, and, if i7 is the
translation
U=Vco&(if>-0),
so that
il,U= (po)' +pa}) (o)' -I- ft)) cos' a + (/> V —pao) {w -- cd) sin' a,
= ^o)' H-^'fi)'' + (y -f ^) 0)0)' cos 2a.
Taking 'sr as the pitch of the resultant screw, and
observing that
o)' sin 2a == ft sin (a -I- 0)y and w sin 2a = ft sin (a — 0),
we obtain the equation,
m sin' 2a = jp' sin' (a + ^) -|-jp8in'(a — 0)
-h(;)'+p)(sin'a-8in'^)cos2a (2).
Again, we obtain, from (1),
z sin 2a = {p' — p) (sin' a — sin' 0\
and if we put y^x tan 0y and eliminate 5, we find that
z(a? + y*) sin 2a = ( jp' — |)) (a:;' sin' a — y* cos' a),
is the Cartesian equation of the surface traced out by the
axis of the resultant screw when p and p' are fixed, and
fiD, 0)' are variable.
88 SCREWS.
103. The shortest distance between the dxes of two screws is
2c, and 2a is the inclination to each other of the axes ; it is
required to find the resultant screw.
Take the middle point of the shortest distance as origin,
and as axes of x and y, take the straight lines bisecting the
angles 2a and tt — 2a.
Let AC he the axis of the screw |>, and take for its
equations
y = — a? tan a, 2; = — c.
Then, if BD is the axis of the screw p\ its equations are
ys=a?tana, z=^c.
Let OK and OL be the projections on the plane xy of
the axes AG and BD.
Applying equal and opposite spins about OK and OL,
the screw p is equivalent to the translation v in OK, the
spin CO about OK and the translation con perpendicular to
OK, in the direction figured. Fig. (66).
Similarly the screw jp' is equivalent to the translations \)'
in OLy the spin to about OL, and the translation ca>' in the
direction perpendicular to OL,
The motion is therefore equivalent to the translations,
parallel to Ox and Oy,
(t;'+ i;)cos a — (w'-f (o)c sin a, (v'— i;)sina +(a)'— a))ccosa. . .(1),
and the spins about Ox and Oy,
(<»' -f ©) cos a, (o)' — (»)sina (2).
As before take XI and V as the resultants, and <^ as the
inclinations to Ox, so that the above quantities are equal to
Fco8<^, F8in<^, flcos^, ft sin ^.
CYUNDROn). 89
If we take z such that
^n = F8m(^-^) (3),
we obtain
2:fl* = (p' "'p) ««' sin 2a + («" - «*) c,
which determines the position of the resultant screw axis.
If 3s 0, we obtain the result of Art. 102.
The spin of the resultant screw is XI, and if TJ is the
translation,
J7=Fcos(<^-^) (4),
leading to
n [7 = jp'©'* + p©' + (p' + jp) «® ' cos 2a — 2cfi>co' sin 2a.
If we put y = a? tan tf, we obtain, from (3),
z{a?'\-%f) sin 2a = (jp' — jp) (a^ sin* a — y* cos* a) + iacy sin 2a.
104. CJonversely, any screw can be decomposed into two
screws having their axes in any two planes parallel to the
axis of the given screw, and having these axes inclined to
each other at any given angle.
These conditions fix the values of a, c, and ^, and, if
we take TJ and Xi as the elements of the given screw, we can
assume at pleasure, so that V and ^ will be determined by
the equations (3) and (4), and then the equivalences of Fcos ^,
Fsin ^, n cos 6, Xi sin Q to the expressions (1) and (2) will
determine the elements of the two screws.
105. To prove iha^ a cylindroid is completely determined
if two screws are given.
Adopting the notation of Art. 103, let axes of f and v,
parallel to the plane ay, meet in a point on the axis of ^ at a
depth f below the plane of xy, and let y^ be the inclination
of the axis of x to the axis of f , so that -^ + a and -^ — a are
the inclinations to the axis of ^ of the axes of the screws p'
and j>.
90 CYLINDROID.
Also let fsr and «r' be the pitches of the f and 17 screws.
Then if the p and p' screws are on the cylindroid defined
by vr and vr', we have from Art. 100,
y = «rcos*('^ + a)+«r'sin"('^ + a) (1),
|) = «r.cos*('^ — a) + «r'sin"('^— a) (2),
^+ c=^(fsr' -fsr)sm {yft + a) cos {'f + a) (3),
f— c = ('ar' — 'or)sin ('^ — a)cos ('^ — a) (4).
From (1) and (2) we obtain
p' —p^ (vr* — -or) sin 2-^ sin 2a,
and from (3) and (4)^
2c = («•' — or) cos 2-^ sin 2a (5) ;
and .*. 2c tan 2-^ =|> — p', which determines '^.
Again p' +p = «r' + «r — («r' — vr) cos 2yjr COS 2a
= «r' + «r — 2ccot2a (6)
and, adding together (3) and (4),
2{'= («r' — -cr) sin 2-^ cos 2a
or 25"= (p' — p) cot 2a, which determines f.
Lastly vr and «r' are given by the equations (5) and (6)^
and the cylindroid is therefore completely determined.
It is obvious that the resultant of the screws p and p' is
on the cylindroid given by vr and vr\
For each screw is decomposable into two having f and v
for axes, and the two pairs of f and 17 components are each
equivalent to one f component and one r) component, and
therefore to a screw on the cylindroid.
It will be seen also that the spins of the three screws
follow the parallelogrammic law and are therefore in the ratio
of the sines of the angles between the axes of the screws.
ROLLING SPHERE. 91
106. The equations of the oasis of the instantafieons
screw.
The motion of a rigid body being completely represented
by the motion of a pomt of the body, and a rotation about
some axis through that point, let u, v, w be the component
velocities of the point (/, and o^, o,, a>, the component
angular velocities about axes through 0\
If x, ify z are coordinates of a point P of the body
referred to (X, its velocities relative to 0' are
and the actual velocities of P are obtained by adding u, v, v/
to these expressions.
The point P will be a point in the screw axis if the
direction of motion of P is coincident with the direction of
the axis of resultant angular velocity, that is, if
Ci), Ci)„ O).
1 2 S
These then are the equations of the screw axis referred
to 0', and, knowing the position of the axes through 0'
relative to axes fixed in space, we can determine the position,
relative to those fixed axes, of the screw axis.
107. A sphere rolls between two parallel surfaces of
revolution which are rotating abov^t their common axis with
different angular velocities; it is required to determine
the motion, and the axodes.
If P and Q are the points of the sphere in contact
with the surfaces at the distances r and / from the axis,
the velocities of P and Q are the same as those of the
points in the surface with which they are in contact, and
are therefore, fig. (67),
(or and ojV
in the direction perpendicular to the plane COE,
92 ROLLING SPHEBE.
Hence if F is the velocity of C,
in the same direction, and it follows that C moves in the
<jircle centre N, with the velocity F.
If the system be started from a state of repose, the only
angular velocity of the sphere about C will be about the axis
CA, and if ft be this angular velocity,
^ _ o) V — ©r
"" ~2^ •
If we take the point F such that
V^il.CF,
the instantaneous axis is the line FE parallel to CA, and
meeting the axis of the surfaces in a fixed point.
Hence the fixed axode is the right circular cone, vertex
E, axis EG, and vertical angle 2FEG, and the moving
axode is the cone, vertex E, axis EC and vertical angle
2FEa
108. A sphere rolls between two parallel planes, which
are rotating, with equal angular velocities in the same
direction, about fixed a^es perpendicular to the planes ; to
determine the motion and the axodes.
Let the plane through G, the centre of the sphere,
parallel to the given planes, intersect the two axes of rota-
tion in A and B, and if P, Q are the points of contact, take
P above this plane, and Q beneath it.
Take c for the radius of the sphere and let
AE^EB^a.
Starting the system by suddenly setting the planes in
motion, the velocities of P parallel to x and y, i.e. perpen-
dicular and parallel to AB, are to, AN and cjCN; and the
velocities of Q are wBN and odGN, N being the projection of
C upon AB.
PARALLEL PLANES. 93
Therefore the velocities of C parallel to x and y are
mEN and w . CN,
that is, the point C moves uniformly in the circle, centre
E and radius EC (b say).
Again there is no angular velocity about Gxy because
the velocities of P and Q in the direction Cy are equal;
but the angular velocity about Cy
o) . AN — wBIf a ^
2^ = -«- = nsay.
The motion is now represented by the translation tob
in CK perpendicular to EC, and the rotation ft about Cy,
The screw axis is parallel to Cy and is at the height
above C given by the equation
z£l 4- a)& cos -^ = 0, or to-z^cob cos yft,
c
the angle KCx being '^, so that
z^-.EK
a
The spin about the screw axis is — cd - , and the translation
is a)b sin '^.
The axodes are clearly cylindrical surfaces, and if CN^x,
we have a? = 6 sin -^j
is the equation of the fixed axode.
To find the moving axode, observe that while EG turns
through the angle '^, a vertical diameter of the sphere turns
through the angle -'^, so that, taking p and e as polar
c
coordinates of the screw axis, referred to C and this moving
diameter, we have
be , J a ,
p = z=— cos ur, and e = - -ur ;
'^ a ^ c ^
be C€
.*. p = — cos — ,
94 ROLLING SPHERE.
is the polar equation of the moving axode. This moving
axode rolls on the elliptic cylinder which is the fixed axode,
and slips backwards and forwards with the variable velocity
iob sin ^.
109. We have assumed in the preceding discussion that
the system was originally at rest, and the planes set in
motion, so that no rotation would be produced about the
diameter perpendicular to the planes. If originally the
sphere was rotating about this diameter, the angular velocity
will remain unchanged, as we know from dynamical con-
siderations.
The eflfect would be that the axodes would retain the
same general forms, but that the axes of these cylinders
would not be parallel to the planes.
110. It may be useful to obtain some of the results
of Art. 108 in a different manner.
Taking Cx and Cy as axes, let the motion be represented
by the velocities w, v, of C, and the rotations coj, ©^ of the
sphere about Cx and Cy.
We have to express the fact that the velocities of the
points P and Q of the sphere are the same as those of the
points of the planes with which they are in contact.
liAC^r, BC^r\ CAN ^6, CBN^if>,
the conditions for the point P give the equations,
u + cojg = rci) cos 0y V — cwj = ro) sin 0,
and, for Q,
u — cOg = r 0) cos <l>i v + co)j = ra> sin ^ ;
we have also,
r sin 5 = r' sin ^, and r' cos ^ — r cos 5 = 2a. ♦
Hence we obtain,
2t^ = 6) (r' cos H- r cos ^), 2t; = o) (/ sin ^ + r sin 0),
CO), = — aco, a)j = ;
PAKALLEL PLANES. '95
.-. y u* + »• = ^ ^/r* + r^ - 2rr' COS (^ - ^) = (»6,
shewing that the motion of C is in the circle of radius 6, and
that the only rotation is about Cy,
111. If in Art. 108 the rotation about the axis through
B is in the opposite direction to that of the rotation about
the axis through A, the eflfect is that the centre moves
uniformly in the straight line CN, and that the sphere
rotates with an angular velocity proportional to the length
of EG about the axis which is inclined to Cx at the same
angle as CK.
The fixed axode is a skew surface of the form,
hz (a?* + y*) H- ojoayy = 0,
a conoidal surface, and it will be seen that the moving axode is
also a skew surface.
MISCELLANEOUS EXAMPLES.
1. A smooth rigid wire bent into a curve turns round a
fixed point in its own plane, and pushes a particle before it
in a straight line. Find the form of the curve and shew
that if it move with uniform angular velocity the particle
moves with uniform velocity.
2. Q, 0, P are fixed points in a rod. Q describes a circle
whose centre is and radius a, C describes a straight line
passing through 0; shew that generally P describes an egg-
shaped oval, whose area is — ,— , and that the radii of
curvature of its ends are
ac^
a(6 + c)±6"
where QC =^b,CP=^ c.
3. If a catenary rolls on a straight line, the envelope
roulettes are involutes of parabolas.
4. An ellipse rolls on a fixed horizontal straight line (the
axis of x). Shew that the locus of the highest point of the
ellipse IS given by the equation,
5. An ellipse rolls on a straight line ; prove that the
difiference between the lengths of the radii of curvature at
corresponding points of the paths traced out by the foci is
constant.
6. A cycloid rolls on an equal cycloid, corresponding
points being in contact ; shew that the locus of the centre
of curvature of the rolling curve at the point of contact is a
trochoid whose generating circle is equal to that of either
cycloid.
MISCELLANEOUS EXAMPLES. 97
7. Prove that the intrinsic equation o f the envelope of
the directrix of a catenary of parameter c rolling on a circle
of radius c will be found by eliminating a between the
equations
8 1 ^ . 5 , 1 + sin a
- = =r tan a sec a + 7 log :i ; — ,
c 2 4^1- sm a'
and ^ = a + tan a.
8. The cardioid r = a (1 — cos 5), rolls on a straight line;
prove that the intrinsic equation of the roulette of the
cusp is
28 = 3a (2<l> - sin 2^),
measuring from the point of contact of the cusp.
Prove also that its Cartesian equation is
15
that its area is -j- Tra*, and that the radius of curvature of
the roulette of the cusp is three times its distance from the
point of contact.
9. Shew that the problem for the determination of the
caustic of a curve for rays proceeding from a point is the
same as that of finding the evolute of the roulette traced out
by the point corresponding to the given point, when an
exactly equal curve is rolled upon the given curve, corre-
sponding points being in contact.
Examine in particular what this becomes in the case of
(i) rays proceeding from the focus of a parabola, (ii) rays
proceeding from a point on a circle.
10. Prove that the curve on which an ellipse must roll
in order that its centre may move in a straight line is given
by the equation y/a = dn a?/6, the modulus being the eccen-
tricity of the ellipse.
B. R. 7
98 MISCELLANEOUS EXAMPLES.
11. The right angle BAG slides so that AB, AG touch
respectively two fixed circles ; if c is the distance between
the centres of these circles, prove that the fixed and moving
centrodes are circles of diameters c and 2c.
If r and r' are the radii of the two circles, and if t and
i are the distances from A to the points of contact, prove that
the radius of curvature of the path of the point A is equal to
tA •
2(^ + 0-^'-^'^
Prove also that, if d is the distance between the points
of contact, and c the distance between the centres of the
circles, this expression is equal to
2d»
*% '
3d« + c''~r*-r
12. If a helix rolls on a straight line which it always
touches, while its axis moves in a plane, any point of the
helix traces a cycloid
13. Two cylinders of different radii are placed on a table
with their axes parallel. A board is placed upon them and
drawn along in a direction perpendicular to the axes of the
cylinders. If there be no slipping prove that the spaces
passed over by the centres of the two cylinders are the same,
1 8
and that each is equal to , where 8 = space traversed by
Z COS ft
the board, and 2a; = angle between board and table.
14. Two equal circular discs of radius c with their planes
parallel are fastened at their centres to a bar, the discs being
inclined to the bar at the angle ft. The two wheels thus
formed being rolled alon^ a plane, prove that the intrinsic
equation of the track of either wheel upon the plane is
,8 sin <^
sm-= -.
c cos a
16. The translations of two points of a rigid body are
given in direction and magnitude, and there is no spin about
the line joining them ; find the screw axis.
MISCELLANEOUS EXAMPLES. 99
16. A sphere rolls between two co-axial cylinders,
which are rotating about their common axis; while one of
them is sliding along its axis ; prove that the path of the
centre is a helix, and that the fixed axode is the surface
generated by the tangent to a helix.
17. Find the fixed and moving axodes in the case of the
steady motion of a top.
18. A sphere rolls between two concentric spheres, which
are rotating about fixed diameters; determine the motion of
its centre.
19. The vertex of the tangent cone to a fixed sphere, centre
G, moves round the circumference of a fixed circle, centre 0,
in the plane perpendicular to 00, the cone having no rotation
about its axis ; find the axodes.
20. A sphere rolls between two parallel planes, which
have different spins about fixed axes perpendicular to their
planes ; determine the motion of its centre.
CAMBBIDOE : PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS.
1
Plate 4.
>
- " r
Plate 5.
E
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a
B
63
E
65
66
* •
• •
67
-i
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